Introduction to Open Data Science - Course Project

About the project

Write a short description about the course and add a link to your GitHub repository here. This is an R Markdown (.Rmd) file so you should use R Markdown syntax.

The text continues here.

I just solved the problem of uploading Rstudio to Github before writing chapter 1, thanks to the help of my classmate, I’m very happy to have solved this trouble!

I’m really looking forward to this course, after having studied QSR I’m really looking forward to continuing my in-depth study of R and analytics, and I also really like Kimmo’s course, so I’ve chosen this course.

The R for Health Data Science book was something I had already come across in the QSR and the second time through I was able to get to grips with some of the necessary commands in R. The parts that were difficult to understand before were now understandable and this was a very necessary process. The link: https://github.com/YuRenn/IODS-project


Describe the work you have done this week and summarize your learning. Data wrangling and analysis

1.read date set

learning2014 <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/learning2014.txt",
                            sep = ",", header = T)
  1. Show a graphical overview of the data and show summaries of the variables in the data.
View(learning2014)
dim(learning2014)
## [1] 166   7
str(learning2014)
## 'data.frame':    166 obs. of  7 variables:
##  $ gender  : chr  "F" "M" "F" "M" ...
##  $ age     : int  53 55 49 53 49 38 50 37 37 42 ...
##  $ attitude: num  3.7 3.1 2.5 3.5 3.7 3.8 3.5 2.9 3.8 2.1 ...
##  $ deep    : num  3.58 2.92 3.5 3.5 3.67 ...
##  $ stra    : num  3.38 2.75 3.62 3.12 3.62 ...
##  $ surf    : num  2.58 3.17 2.25 2.25 2.83 ...
##  $ points  : int  25 12 24 10 22 21 21 31 24 26 ...

This data set is from international survey of Approaches to Learning, made possible by Teachers’ Academy funding for KV in 2013-2015. Before data wrangling, the data set has 183 observations and 60 variables. Then I did a data wrangling, and choose the variable of attitude, which is a sum of 10 questions related to students attitude towards statistics. In the dataset of learning2014, it has 166 observations and 7 variables, they are gender, age,attitude(Global attitude toward statistics), deep(deep learning), stra(strategical learning), surf(surface learning), and points(exam points).

3.Choose three variables as explanatory variables and fit a regression model where exam points is the target (dependent, outcome) variable. 3.1First, visualize the data set and create a plot matrix with ggpairs()

library(GGally)
## Loading required package: ggplot2
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
library(ggplot2)
ggpairs(learning2014, lower = list(combo = wrap("facethist", bins = 20)))

3.2Choose attitude, points and surf as variables and fit a regression model

fit <- learning2014 %>%
  lm(points ~ attitude + surf, data = .)
summary(fit)
## 
## Call:
## lm(formula = points ~ attitude + surf, data = .)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -17.277  -3.236   0.386   3.977  10.642 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  14.1196     3.1271   4.515 1.21e-05 ***
## attitude      3.4264     0.5764   5.944 1.63e-08 ***
## surf         -0.7790     0.7956  -0.979    0.329    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.32 on 163 degrees of freedom
## Multiple R-squared:  0.1953, Adjusted R-squared:  0.1854 
## F-statistic: 19.78 on 2 and 163 DF,  p-value: 2.041e-08

The result shows that expect surf, the p-values for the other variables are both significant and predictive. adjust R-squared shows that the model explains 18.54%.

3.3 remove the variable of surf, and fit the model again.

fit2 <- learning2014 %>%
  lm(points ~ attitude, data = .)
summary(fit2)
## 
## Call:
## lm(formula = points ~ attitude, data = .)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.9763  -3.2119   0.4339   4.1534  10.6645 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  11.6372     1.8303   6.358 1.95e-09 ***
## attitude      3.5255     0.5674   6.214 4.12e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared:  0.1906, Adjusted R-squared:  0.1856 
## F-statistic: 38.61 on 1 and 164 DF,  p-value: 4.119e-09

There is a strong correlation between the two variables and the model is a valid model. Adjusted R-squared indicates that the model can explain 18.54 of the variance, which is only 0.01% higher than fit1. A coefficient of 3.53 for attitude means that for every 1 increase in attitude, the test points will increase by 3.53.

4.Model diagnostic

library(ggfortify)
autoplot(fit2)

The Residuals vs fitted plot shows the data distributed up and down along the blue line with no clear trend, indicating a linear relationship.


prepare some packages

library(tidyverse)
## ── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
## ✔ tibble  3.1.8      ✔ dplyr   1.0.10
## ✔ tidyr   1.2.1      ✔ stringr 1.4.1 
## ✔ readr   2.1.3      ✔ forcats 0.5.2 
## ✔ purrr   0.3.4      
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()
library(dplyr)
  1. read the joined student alcohol consumption data
alc <- read_csv("alc.csv")
## Rows: 370 Columns: 35
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (17): school, sex, address, famsize, Pstatus, Mjob, Fjob, reason, guardi...
## dbl (17): age, Medu, Fedu, traveltime, studytime, famrel, freetime, goout, D...
## lgl  (1): high_use
## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

1.2 Print out the names of the variables in the data and describe the data set briefly

glimpse(alc)
## Rows: 370
## Columns: 35
## $ school     <chr> "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP",…
## $ sex        <chr> "F", "F", "F", "F", "F", "M", "M", "F", "M", "M", "F", "F",…
## $ age        <dbl> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15, 15, 15,…
## $ address    <chr> "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U", "U",…
## $ famsize    <chr> "GT3", "GT3", "LE3", "GT3", "GT3", "LE3", "LE3", "GT3", "LE…
## $ Pstatus    <chr> "A", "T", "T", "T", "T", "T", "T", "A", "A", "T", "T", "T",…
## $ Medu       <dbl> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, 3, 3, 4,…
## $ Fedu       <dbl> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, 3, 2, 3,…
## $ Mjob       <chr> "at_home", "at_home", "at_home", "health", "other", "servic…
## $ Fjob       <chr> "teacher", "other", "other", "services", "other", "other", …
## $ reason     <chr> "course", "course", "other", "home", "home", "reputation", …
## $ guardian   <chr> "mother", "father", "mother", "mother", "father", "mother",…
## $ traveltime <dbl> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1,…
## $ studytime  <dbl> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, 2, 1, 1,…
## $ schoolsup  <chr> "yes", "no", "yes", "no", "no", "no", "no", "yes", "no", "n…
## $ famsup     <chr> "no", "yes", "no", "yes", "yes", "yes", "no", "yes", "yes",…
## $ activities <chr> "no", "no", "no", "yes", "no", "yes", "no", "no", "no", "ye…
## $ nursery    <chr> "yes", "no", "yes", "yes", "yes", "yes", "yes", "yes", "yes…
## $ higher     <chr> "yes", "yes", "yes", "yes", "yes", "yes", "yes", "yes", "ye…
## $ internet   <chr> "no", "yes", "yes", "yes", "no", "yes", "yes", "no", "yes",…
## $ romantic   <chr> "no", "no", "no", "yes", "no", "no", "no", "no", "no", "no"…
## $ famrel     <dbl> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, 5, 5, 3,…
## $ freetime   <dbl> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, 3, 5, 1,…
## $ goout      <dbl> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, 2, 5, 3,…
## $ Dalc       <dbl> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1,…
## $ Walc       <dbl> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1, 4, 3,…
## $ health     <dbl> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, 4, 5, 5,…
## $ failures   <dbl> 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0,…
## $ paid       <chr> "no", "no", "yes", "yes", "yes", "yes", "no", "no", "yes", …
## $ absences   <dbl> 5, 3, 8, 1, 2, 8, 0, 4, 0, 0, 1, 2, 1, 1, 0, 5, 8, 3, 9, 5,…
## $ G1         <dbl> 2, 7, 10, 14, 8, 14, 12, 8, 16, 13, 12, 10, 13, 11, 14, 16,…
## $ G2         <dbl> 8, 8, 10, 14, 12, 14, 12, 9, 17, 14, 11, 12, 14, 11, 15, 16…
## $ G3         <dbl> 8, 8, 11, 14, 12, 14, 12, 10, 18, 14, 12, 12, 13, 12, 16, 1…
## $ alc_use    <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1.5, 1.0,…
## $ high_use   <lgl> FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE, FALS…

The dataset has 370 observations and 35 variables. I will explain some meanings of variables: famsize-family size; Pstatus - parent’s cohabitation status; Medu - mother’s education; Fedu - father’s education; Mjob - mother’s job; Fjob - father’s job; reason - reason to choose this school; guardian - student’s guardian; schoolsup - extra educational support; famsup - family educational support; activities - extra-curricular activities; nursery - attended nursery school; higher - wants to take higher education; internet - Internet access at home; romantic - with a romantic relationship; famrel - quality of family relationships; freetime - free time after school; goout - going out with friends; Dalc - workday alcohol consumption; Walc - weekend alcohol consumption; health - current health status; absences - number of school absences; G1 - first period grade; G2 - second period grade; G3 - final grade.

  1. Choose 4 variables and present personal hypothesis The four variables I have chosen are goout - going out with friends, freetime - free time after school,famrel - quality of family relationships and absences - number of school absences. I suspect that these four variables would be related to student drinking; having free time after school and going out with friends multiple times may increase the frequency of drinking, the quality of family relationships may also influence drinking status, and there may be a relationship between absence from class and drinking.

3.Explore the distributions of variables and their relationships with alcohol consumption

alc %>% group_by(freetime) %>% summarise(count = n())
## # A tibble: 5 × 2
##   freetime count
##      <dbl> <int>
## 1        1    17
## 2        2    60
## 3        3   152
## 4        4   105
## 5        5    36
alc %>% group_by(goout) %>% summarise(count = n())
## # A tibble: 5 × 2
##   goout count
##   <dbl> <int>
## 1     1    22
## 2     2    97
## 3     3   120
## 4     4    78
## 5     5    53
alc %>% group_by(famrel) %>% summarise(count = n())
## # A tibble: 5 × 2
##   famrel count
##    <dbl> <int>
## 1      1     8
## 2      2    18
## 3      3    64
## 4      4   180
## 5      5   100
alc %>% group_by(absences) %>% summarise(count = n())
## # A tibble: 26 × 2
##    absences count
##       <dbl> <int>
##  1        0    63
##  2        1    50
##  3        2    56
##  4        3    38
##  5        4    35
##  6        5    22
##  7        6    21
##  8        7    12
##  9        8    20
## 10        9    11
## # … with 16 more rows

The distribution of the freetime and goout variables is more in the middle, with the family relationship variables mostly distributed between 3 and 5, indicating that the majority of students have good family relationships; classroom absences, the majority of students are absent 5 times or less.

3.1 explore alcohol high use and freetime

p1 <- alc %>%
  ggplot(aes(x = freetime, fill = high_use)) +
  geom_bar(position = "fill", color ="white") +
  labs(x = "free time after school", y = "alcohol high-user",
       title = "Proportion of alcohol high-use by freetime")+
  theme(legend.position = "bottom")+
  scale_fill_discrete(labels = c("FALSE" = "Non-high-user", 
                                 "TRUE" = "high-user"))+
  scale_fill_brewer(palette = "")
## Warning in pal_name(palette, type): Unknown palette
## Scale for 'fill' is already present. Adding another scale for 'fill', which
## will replace the existing scale.
p1

According to the proportions shown in the bar chart, the proportion of alcohol use increases gradually as leisure time after school increases.

3.2 explore alcohol high use and goout

p2 <- alc %>%
  ggplot(aes(x = goout, fill = high_use)) +
  geom_bar(position = "fill", color ="white") +
  labs(x = "going out with friends", y = "alcohol high-user",
       title = "Proportion of alcohol high-use by goout")+
  theme(legend.position = "bottom")+
  scale_fill_discrete(labels = c("FALSE" = "Non-high-user", 
                                 "TRUE" = "high-user"))+
  scale_fill_brewer(palette = "")
## Warning in pal_name(palette, type): Unknown palette
## Scale for 'fill' is already present. Adding another scale for 'fill', which
## will replace the existing scale.
p2

According to the bar scale, the proportion of alcohol consumption increases as the number of outings with friends increases, particularly at level 4 of the outing degree, where the proportion increases substantially.

3.3explore alcohol high use and famrel

p3 <- alc %>%
  ggplot(aes(x = famrel, fill = high_use)) +
  geom_bar(position = "fill", color ="white") +
  labs(x = "quality of family relationships", y = "alcohol high-user",
       title = "Proportion of alcohol high-use by family relationship")+
  theme(legend.position = "bottom")+
  scale_fill_discrete(labels = c("FALSE" = "Non-high-user", 
                                 "TRUE" = "high-user"))+
  scale_fill_brewer(palette = "")
## Warning in pal_name(palette, type): Unknown palette
## Scale for 'fill' is already present. Adding another scale for 'fill', which
## will replace the existing scale.
p3

“famrel - quality of family relationships (numeric: from 1 - very bad to 5 - excellent)” Family relationships start at level 3, with better family relationships being accompanied by a lower proportion of alcohol consumption. The highest proportion of alcohol consumption is found when family relationships are poor.

3.4 explore alcohol high use and absences

p4 <- alc %>%
  ggplot(aes(x = absences, fill = high_use)) +
  geom_bar(position = "fill", color ="white") +
  labs(x = " number of school absences (numeric: from 0 to 93)", y = "alcohol high-user",
       title = "Proportion of alcohol high-use by number of school absences")+
  theme(legend.position = "bottom")+
  scale_fill_discrete(labels = c("FALSE" = "Non-high-user", 
                                 "TRUE" = "high-user"))+
  scale_fill_brewer(palette = "")
## Warning in pal_name(palette, type): Unknown palette
## Scale for 'fill' is already present. Adding another scale for 'fill', which
## will replace the existing scale.
p4

As the number of absences from school increases, the proportion of alcohol consumption increases with it. The bars show much the same results as I expected, the only differences being family relationships and alcohol consumption rates, with the worst families not having the highest rates of alcohol consumption and the second worst families having the highest rates of alcohol consumption.

4.logistic regression model 4.1

m1 <- glm(high_use ~ freetime + goout + famrel + absences, data = alc, family = "binomial")

summary(m1)
## 
## Call:
## glm(formula = high_use ~ freetime + goout + famrel + absences, 
##     family = "binomial", data = alc)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -1.8393  -0.7676  -0.5200   0.9080   2.3909  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  -2.7391     0.6975  -3.927  8.6e-05 ***
## freetime      0.2283     0.1393   1.639 0.101292    
## goout         0.7103     0.1252   5.673  1.4e-08 ***
## famrel       -0.4019     0.1387  -2.898 0.003754 ** 
## absences      0.0761     0.0220   3.459 0.000543 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 452.04  on 369  degrees of freedom
## Residual deviance: 380.08  on 365  degrees of freedom
## AIC: 390.08
## 
## Number of Fisher Scoring iterations: 4

There is a highly significant relationship between the proportion of increased drinking and family relationships, the frequency of going out with friends and absence from class, but a non-significant p-value between this and free time.

4.2 From coefficients to odds ratios

m1 <- glm(high_use ~ freetime + goout + famrel + absences, data = alc, family = "binomial")

# compute odds ratios (OR)
OR <- coef(m1) %>% exp
OR
## (Intercept)    freetime       goout      famrel    absences 
##  0.06462506  1.25649062  2.03450512  0.66907685  1.07907060
# compute confidence intervals (CI)
confint(m1)
## Waiting for profiling to be done...
##                   2.5 %     97.5 %
## (Intercept) -4.14545349 -1.4024140
## freetime    -0.04305995  0.5046433
## goout        0.47111595  0.9630772
## famrel      -0.67730666 -0.1316918
## absences     0.03387575  0.1213892
CI <- exp(confint(m1))
## Waiting for profiling to be done...
CI
##                  2.5 %    97.5 %
## (Intercept) 0.01583625 0.2460024
## freetime    0.95785396 1.6563946
## goout       1.60178070 2.6197457
## famrel      0.50798332 0.8766111
## absences    1.03445607 1.1290642
# print out the odds ratios with their confidence intervals
cbind(OR, CI)
##                     OR      2.5 %    97.5 %
## (Intercept) 0.06462506 0.01583625 0.2460024
## freetime    1.25649062 0.95785396 1.6563946
## goout       2.03450512 1.60178070 2.6197457
## famrel      0.66907685 0.50798332 0.8766111
## absences    1.07907060 1.03445607 1.1290642

According to the model, the probability of drinking increases 1.26 times for every doubling of free time, 2.03 times for more frequent outings with friends, 0.67 times for every unit of improvement in family relations, and 1.08 times for every increase in absenteeism from school.

  1. Explore the predictive power of the model 5.1 Binary predictions(1)
prob <- predict(m1, type = "response")
library(dplyr)

alc <- mutate(alc, probability = prob)
alc <- mutate(alc, prediction = probability > 0.5)

table(high_use = alc$high_use, prediction = alc$prediction)
##         prediction
## high_use FALSE TRUE
##    FALSE   239   20
##    TRUE     67   44

Of the 370 participants, there are 259 non-alcoholics, of which the model correctly predicted 239 (92.28%); of the 111 alcoholics, the model successfully predicted 44 (39.64%).

5.2 Binary predictions (2)

library(dplyr); library(ggplot2)

# initialize a plot of 'high_use' versus 'probabilities' in 'alc'
g <- ggplot(alc, aes(x = probability, y = high_use))

# define the geom as points and draw the plot
g+aes(color = prediction, shape = prediction) + geom_point()

# tabulate the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$prediction)
##         prediction
## high_use FALSE TRUE
##    FALSE   239   20
##    TRUE     67   44

6.Bonus

# define a loss function (mean prediction error)
loss_func <- function(class, prob) {
  n_wrong <- abs(class - prob) > 0.5
  mean(n_wrong)
}
training.error.full <- loss_func(alc$high_use, alc$probability)
training.error.full
## [1] 0.2351351

Prediction error rate of 23.51%.


  1. Load the Boston data from the MASS package
library(MASS)
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
## 
##     select
library(corrplot)
## corrplot 0.92 loaded
library(tidyverse)
data("Boston")

1,1 explore the structure and the dimensions of the data

#explore the structure
str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
#explore the dimensions#
dim(Boston)
## [1] 506  14

This dataset has 506 observations and 14 variables. And the variable crim = per capita crime rate by town; zn = proportion of residential land zoned for lots over 25,000 sq.ft; indus = proportion of non-retail business acres per town; chas = Charles River dummy variable (= 1 if tract bounds river; 0 otherwise); nox = nitrogen oxides concentration (parts per 10 million); rm = average number of rooms per dwelling; dis = weighted mean of distances to five Boston employment centres; rad = index of accessibility to radial highways; tax = full-value property-tax rate per $10,000; ptratio = pupil-teacher ratio by town; black = 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town; lstat = lower status of the population (percent); medv = median value of owner-occupied homes in $1000s.

2.Show a graphical overview of the data and show summaries of the variables in the data.

# plot matrix of the variables
pairs(Boston)

# calculate the correlation matrix and round it
cor_matrix <- cor(Boston) 

# print the correlation matrix
cor_matrix
##                crim          zn       indus         chas         nox
## crim     1.00000000 -0.20046922  0.40658341 -0.055891582  0.42097171
## zn      -0.20046922  1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus    0.40658341 -0.53382819  1.00000000  0.062938027  0.76365145
## chas    -0.05589158 -0.04269672  0.06293803  1.000000000  0.09120281
## nox      0.42097171 -0.51660371  0.76365145  0.091202807  1.00000000
## rm      -0.21924670  0.31199059 -0.39167585  0.091251225 -0.30218819
## age      0.35273425 -0.56953734  0.64477851  0.086517774  0.73147010
## dis     -0.37967009  0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad      0.62550515 -0.31194783  0.59512927 -0.007368241  0.61144056
## tax      0.58276431 -0.31456332  0.72076018 -0.035586518  0.66802320
## ptratio  0.28994558 -0.39167855  0.38324756 -0.121515174  0.18893268
## black   -0.38506394  0.17552032 -0.35697654  0.048788485 -0.38005064
## lstat    0.45562148 -0.41299457  0.60379972 -0.053929298  0.59087892
## medv    -0.38830461  0.36044534 -0.48372516  0.175260177 -0.42732077
##                  rm         age         dis          rad         tax    ptratio
## crim    -0.21924670  0.35273425 -0.37967009  0.625505145  0.58276431  0.2899456
## zn       0.31199059 -0.56953734  0.66440822 -0.311947826 -0.31456332 -0.3916785
## indus   -0.39167585  0.64477851 -0.70802699  0.595129275  0.72076018  0.3832476
## chas     0.09125123  0.08651777 -0.09917578 -0.007368241 -0.03558652 -0.1215152
## nox     -0.30218819  0.73147010 -0.76923011  0.611440563  0.66802320  0.1889327
## rm       1.00000000 -0.24026493  0.20524621 -0.209846668 -0.29204783 -0.3555015
## age     -0.24026493  1.00000000 -0.74788054  0.456022452  0.50645559  0.2615150
## dis      0.20524621 -0.74788054  1.00000000 -0.494587930 -0.53443158 -0.2324705
## rad     -0.20984667  0.45602245 -0.49458793  1.000000000  0.91022819  0.4647412
## tax     -0.29204783  0.50645559 -0.53443158  0.910228189  1.00000000  0.4608530
## ptratio -0.35550149  0.26151501 -0.23247054  0.464741179  0.46085304  1.0000000
## black    0.12806864 -0.27353398  0.29151167 -0.444412816 -0.44180801 -0.1773833
## lstat   -0.61380827  0.60233853 -0.49699583  0.488676335  0.54399341  0.3740443
## medv     0.69535995 -0.37695457  0.24992873 -0.381626231 -0.46853593 -0.5077867
##               black      lstat       medv
## crim    -0.38506394  0.4556215 -0.3883046
## zn       0.17552032 -0.4129946  0.3604453
## indus   -0.35697654  0.6037997 -0.4837252
## chas     0.04878848 -0.0539293  0.1752602
## nox     -0.38005064  0.5908789 -0.4273208
## rm       0.12806864 -0.6138083  0.6953599
## age     -0.27353398  0.6023385 -0.3769546
## dis      0.29151167 -0.4969958  0.2499287
## rad     -0.44441282  0.4886763 -0.3816262
## tax     -0.44180801  0.5439934 -0.4685359
## ptratio -0.17738330  0.3740443 -0.5077867
## black    1.00000000 -0.3660869  0.3334608
## lstat   -0.36608690  1.0000000 -0.7376627
## medv     0.33346082 -0.7376627  1.0000000
# visualize the correlation matrix
library(corrplot)
corrplot(cor_matrix, method="circle")

From the correlation matrix, it is possible to look for relationships between different variables, using the variable dis as an example, the weighted average of distances to the five employment centres in Boston and the variable indus (proportion of non-retail commercial acres in each town) show a very strong negative correlation; with nox (nitrogen oxide concentration) also a strong negative correlation; and with age again a strong negative correlation. There is a relatively strong positive correlation with zn (the proportion of residential land over 25,000 square feet).

#summaries of the variables in the data
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08205   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00
  1. Standardize the dataset and print out summaries of the scaled data.
# center and standardize variables
boston_scaled <- scale(Boston)

# summaries of the scaled variables
summary(boston_scaled)
##       crim                 zn               indus              chas        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563   Min.   :-0.2723  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668   1st Qu.:-0.2723  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109   Median :-0.2723  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150   3rd Qu.:-0.2723  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202   Max.   : 3.6648  
##       nox                rm               age               dis         
##  Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331   Min.   :-1.2658  
##  1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366   1st Qu.:-0.8049  
##  Median :-0.1441   Median :-0.1084   Median : 0.3171   Median :-0.2790  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059   3rd Qu.: 0.6617  
##  Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164   Max.   : 3.9566  
##       rad               tax             ptratio            black        
##  Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047   Min.   :-3.9033  
##  1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876   1st Qu.: 0.2049  
##  Median :-0.5225   Median :-0.4642   Median : 0.2746   Median : 0.3808  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.4332  
##  Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372   Max.   : 0.4406  
##      lstat              medv        
##  Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 3.5453   Max.   : 2.9865
# class of the boston_scaled object
class(boston_scaled)
## [1] "matrix" "array"
# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)
boston_scaled
##             crim          zn       indus       chas         nox           rm
## 1   -0.419366929  0.28454827 -1.28663623 -0.2723291 -0.14407485  0.413262920
## 2   -0.416926670 -0.48724019 -0.59279438 -0.2723291 -0.73953036  0.194082387
## 3   -0.416928995 -0.48724019 -0.59279438 -0.2723291 -0.73953036  1.281445551
## 4   -0.416338404 -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.015297761
## 5   -0.412074053 -0.48724019 -1.30558569 -0.2723291 -0.83445805  1.227362043
## 6   -0.416631374 -0.48724019 -1.30558569 -0.2723291 -0.83445805  0.206891639
## 7   -0.409837246  0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.388026950
## 8   -0.403296561  0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.160306916
## 9   -0.395543302  0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.930285282
## 10  -0.400333140  0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.399412952
## 11  -0.393956378  0.04872402 -0.47618230 -0.2723291 -0.26489191  0.131459378
## 12  -0.406444832  0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.392296701
## 13  -0.409198989  0.04872402 -0.47618230 -0.2723291 -0.26489191 -0.563086727
## 14  -0.346886928 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.477691714
## 15  -0.345933611 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 16  -0.347162460 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.641365488
## 17  -0.297573695 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.497617217
## 18  -0.328932014 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.419338455
## 19  -0.326780075 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.179354069
## 20  -0.335721492 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.793653261
## 21  -0.274570851 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -1.017103545
## 22  -0.321045059 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.454919710
## 23  -0.276816959 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.203004422
## 24  -0.305188606 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 25  -0.332877817 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.513272969
## 26  -0.322382028 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.975829289
## 27  -0.341986646 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.671253743
## 28  -0.308985598 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.338213193
## 29  -0.330235268 -0.48724019 -0.43682573 -0.2723291 -0.14407485  0.299402903
## 30  -0.303558666 -0.48724019 -0.43682573 -0.2723291 -0.14407485  0.554164692
## 31  -0.288635766 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.813578764
## 32  -0.262604396 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.302631937
## 33  -0.258736486 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.476268464
## 34  -0.286204807 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.830657767
## 35  -0.232598159 -0.48724019 -0.43682573 -0.2723291 -0.14407485 -0.268473932
## 36  -0.412641393 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.500463717
## 37  -0.408773484 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.631402737
## 38  -0.410784750 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.618593485
## 39  -0.399750686 -0.48724019 -0.75459363 -0.2723291 -0.48063666 -0.453496460
## 40  -0.416889467  2.72854505 -1.19334657 -0.2723291 -1.09335175  0.441727925
## 41  -0.416196569  2.72854505 -1.19334657 -0.2723291 -1.09335175  1.052302267
## 42  -0.405285738 -0.48724019 -0.61611679 -0.2723291 -0.92075595  0.690796712
## 43  -0.403651148 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.164576667
## 44  -0.401574777 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.104800158
## 45  -0.405837965 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.306901688
## 46  -0.400172703 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.857699521
## 47  -0.398203290 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.709681499
## 48  -0.393447167 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.362408446
## 49  -0.390587216 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -1.260479332
## 50  -0.394551620 -0.48724019 -0.61611679 -0.2723291 -0.92075595 -0.971559538
## 51  -0.409786092  0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.457766211
## 52  -0.415059564  0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.241432178
## 53  -0.413870242  0.41317968 -0.80123846 -0.2723291 -0.99842406  0.322174907
## 54  -0.414310861  0.41317968 -0.80123846 -0.2723291 -0.99842406 -0.407952453
## 55  -0.418520570  2.72854505 -1.04029322 -0.2723291 -1.24868797 -0.564509977
## 56  -0.418577536  3.37170210 -1.44552019 -0.2723291 -1.30909650  1.372533565
## 57  -0.417712575  3.15731641 -1.51548743 -0.2723291 -1.24868797  0.139998879
## 58  -0.418436864  3.80047346 -1.43094368 -0.2723291 -1.24005818  0.756266222
## 59  -0.402145605  0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.198734672
## 60  -0.408094536  0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.509003218
## 61  -0.402742009  0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.773727758
## 62  -0.400138988  0.58468822 -0.87557866 -0.2723291 -0.87760700 -0.453496460
## 63  -0.407281891  0.58468822 -0.87557866 -0.2723291 -0.87760700  0.243896145
## 64  -0.405395021  0.58468822 -0.87557866 -0.2723291 -0.87760700  0.679410711
## 65  -0.417833484  0.26310970 -1.42219777 -0.2723291 -1.19604625  1.166162284
## 66  -0.415934988  2.94293073 -1.13212523 -0.2723291 -1.35224545  0.007636609
## 67  -0.415010735  2.94293073 -1.13212523 -0.2723291 -1.35224545 -0.708258248
## 68  -0.413371495  0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.578742479
## 69  -0.404344047  0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.982945540
## 70  -0.405202032  0.04872402 -0.73855947 -0.2723291 -1.25731776 -0.568779727
## 71  -0.409840734 -0.48724019 -0.04763292 -0.2723291 -1.22279860  0.188389387
## 72  -0.401644532 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.460612711
## 73  -0.409447781 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.312594689
## 74  -0.397385995 -0.48724019 -0.04763292 -0.2723291 -1.22279860 -0.056409650
## 75  -0.410921935 -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.016558644
## 76  -0.409043203 -0.48724019  0.24681257 -0.2723291 -1.01568364  0.001943608
## 77  -0.408297988 -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.008019143
## 78  -0.409979081 -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.205850923
## 79  -0.413537744 -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.074911903
## 80  -0.410351107 -0.48724019  0.24681257 -0.2723291 -1.01568364 -0.584435480
## 81  -0.415319982  0.58468822 -0.91493524 -0.2723291 -1.11061133  0.629596953
## 82  -0.414914241  0.58468822 -0.91493524 -0.2723291 -1.11061133  0.475885930
## 83  -0.415847794  0.58468822 -0.91493524 -0.2723291 -1.11061133  0.024715612
## 84  -0.415973353  0.58468822 -0.91493524 -0.2723291 -1.11061133 -0.167423167
## 85  -0.414220179 -0.48724019 -0.96886832 -0.2723291 -0.91212616  0.148538381
## 86  -0.413434274 -0.48724019 -0.96886832 -0.2723291 -0.91212616  0.491541682
## 87  -0.414070206 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.383757200
## 88  -0.411788058 -0.48724019 -0.96886832 -0.2723291 -0.91212616 -0.232892677
## 89  -0.413521468 -0.48724019 -1.12629463 -0.2723291 -0.56693456  1.028107013
## 90  -0.413937672 -0.48724019 -1.12629463 -0.2723291 -0.56693456  1.130581029
## 91  -0.414656148 -0.48724019 -1.12629463 -0.2723291 -0.56693456  0.188389387
## 92  -0.415530409 -0.48724019 -1.12629463 -0.2723291 -0.56693456  0.171310384
## 93  -0.415215350  0.71331963  0.56895343 -0.2723291 -0.78267931  0.223970642
## 94  -0.416759258  0.71331963  0.56895343 -0.2723291 -0.78267931 -0.104800158
## 95  -0.415109555  0.71331963  0.56895343 -0.2723291 -0.78267931 -0.050716649
## 96  -0.405913532 -0.48724019 -1.20209248 -0.2723291 -0.94664532  0.484425431
## 97  -0.406727340 -0.48724019 -1.20209248 -0.2723291 -0.94664532 -0.173116168
## 98  -0.406054205 -0.48724019 -1.20209248 -0.2723291 -0.94664532  2.539598741
## 99  -0.410583624 -0.48724019 -1.20209248 -0.2723291 -0.94664532  2.185209437
## 100 -0.412126370 -0.48724019 -1.20209248 -0.2723291 -0.94664532  1.610216351
## 101 -0.402818740 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.629596953
## 102 -0.406811046 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.706452465
## 103 -0.393506459 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.171310384
## 104 -0.395500287 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.210120673
## 105 -0.403872039 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.167423167
## 106 -0.404683521 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.617170235
## 107 -0.400198280 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.638518988
## 108 -0.404852095 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.224353176
## 109 -0.405218308 -0.48724019 -0.37560439 -0.2723291 -0.29941107  0.269514649
## 110 -0.389452536 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.079181654
## 111 -0.407553935 -0.48724019 -0.37560439 -0.2723291 -0.29941107 -0.127572161
## 112 -0.408378206 -0.48724019 -0.16424500 -0.2723291 -0.06640675  0.612517950
## 113 -0.405768210 -0.48724019 -0.16424500 -0.2723291 -0.06640675 -0.528928721
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## 377  1.357253412 -0.48724019  1.01499462 -0.2723291  1.00368721  0.518583436
## 378  0.721959412 -0.48724019  1.01499462 -0.2723291  1.00368721  0.724954717
## 379  2.329195069 -0.48724019  1.01499462 -0.2723291  1.00368721  0.135729129
## 380  1.657048387 -0.48724019  1.01499462 -0.2723291  1.00368721 -0.087721155
## 381  9.924109610 -0.48724019  1.01499462 -0.2723291  1.00368721  0.972600255
## 382  1.425427210 -0.48724019  1.01499462 -0.2723291  1.00368721  0.370565414
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## 384  0.509089517 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.088266056
## 385  1.914932287 -0.48724019  1.01499462 -0.2723291  1.25395112 -2.727850303
## 386  1.534407630 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.434115858
## 387  2.415877170 -0.48724019  1.01499462 -0.2723291  1.25395112 -2.323647242
## 388  2.206996093 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.828356167
## 389  1.246308229 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.999146193
## 390  0.527604795 -0.48724019  1.01499462 -0.2723291  1.25395112 -1.273288584
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## 396  0.593291831 -0.48724019  1.01499462 -0.2723291  1.19354259  0.265244898
## 397  0.262572179 -0.48724019  1.01499462 -0.2723291  1.19354259  0.171310384
## 398  0.471833420 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.765188257
## 399  4.038608880 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.183623820
## 400  0.732778398 -0.48724019  1.01499462 -0.2723291  1.19354259 -0.615746985
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## 402  1.234973056 -0.48724019  1.01499462 -0.2723291  1.19354259  0.083068871
## 403  0.695478123 -0.48724019  1.01499462 -0.2723291  1.19354259  0.169887134
## 404  2.463298882 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.331641842
## 405  4.408007629 -0.48724019  1.01499462 -0.2723291  1.19354259 -1.072610303
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## 407  1.988326078 -0.48724019  1.01499462 -0.2723291  0.90012973 -3.055197852
## 408  0.969311483 -0.48724019  1.01499462 -0.2723291  0.90012973 -0.963020037
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## 410  1.258468834 -0.48724019  1.01499462 -0.2723291  0.36508275  0.807503230
## 411  5.524853484 -0.48724019  1.01499462 -0.2723291  0.36508275 -0.750955755
## 412  1.213407163 -0.48724019  1.01499462 -0.2723291  0.36508275  0.529969438
## 413  1.766830989 -0.48724019  1.01499462 -0.2723291  0.36508275 -2.357805247
## 414  2.911369543 -0.48724019  1.01499462 -0.2723291  0.36508275 -1.607752384
## 415  4.898256758 -0.48724019  1.01499462 -0.2723291  1.19354259 -2.512939520
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## 417  0.839462719 -0.48724019  1.01499462 -0.2723291  1.07272553  0.707875715
## 418  2.595705326 -0.48724019  1.01499462 -0.2723291  1.07272553 -1.395688102
## 419  8.128839131 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.466305712
## 420  0.953174847 -0.48724019  1.01499462 -0.2723291  1.40928733  0.767652224
## 421  0.868899291 -0.48724019  1.01499462 -0.2723291  1.40928733  0.179849885
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## 426  1.423787970 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.553123975
## 427  1.003735531 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.637095738
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## 429  0.436385137 -0.48724019  1.01499462 -0.2723291  1.07272553 -0.130418661
## 430  0.665620696 -0.48724019  1.01499462 -0.2723291  1.07272553  0.135729129
## 431  0.567177918 -0.48724019  1.01499462 -0.2723291  0.25289548  0.090185122
## 432  0.749723028 -0.48724019  1.01499462 -0.2723291  0.25289548  0.780461476
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## 437  1.256434316 -0.48724019  1.01499462 -0.2723291  1.59914271  0.251012396
## 438  1.344372005 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.188771920
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## 441  2.143519126 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.664137492
## 442  0.710413811 -0.48724019  1.01499462 -0.2723291  1.59914271  0.172733634
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## 444  0.738590145 -0.48724019  1.01499462 -0.2723291  1.59914271  0.285170401
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## 446  0.820582390 -0.48724019  1.01499462 -0.2723291  1.59914271  0.248165896
## 447  0.310937908 -0.48724019  1.01499462 -0.2723291  1.59914271  0.080222370
## 448  0.733743341 -0.48724019  1.01499462 -0.2723291  1.59914271 -0.047870149
## 449  0.664481366 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.141804663
## 450  0.454858563 -0.48724019  1.01499462 -0.2723291  1.36613839  0.188389387
## 451  0.360888236 -0.48724019  1.01499462 -0.2723291  1.36613839  0.660908458
## 452  0.212475366 -0.48724019  1.01499462 -0.2723291  1.36613839  0.527122938
## 453  0.171672232 -0.48724019  1.01499462 -0.2723291  1.36613839  0.017599361
## 454  0.538806271 -0.48724019  1.01499462 -0.2723291  1.36613839  1.577481596
## 455  0.685935651 -0.48724019  1.01499462 -0.2723291  1.36613839  0.631020203
## 456  0.132400217 -0.48724019  1.01499462 -0.2723291  1.36613839  0.342100410
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## 458  0.533282845 -0.48724019  1.01499462 -0.2723291  1.36613839 -0.496193967
## 459  0.481158489 -0.48724019  1.01499462 -0.2723291  1.36613839  0.023292362
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## 461  0.139347806 -0.48724019  1.01499462 -0.2723291  1.36613839  0.592592447
## 462  0.009252575 -0.48724019  1.01499462 -0.2723291  1.36613839  0.130036128
## 463  0.353587222 -0.48724019  1.01499462 -0.2723291  1.36613839  0.046064365
## 464  0.256654638 -0.48724019  1.01499462 -0.2723291  1.36613839  0.325021407
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## 468  0.094024554 -0.48724019  1.01499462 -0.2723291  0.25289548 -0.400836202
## 469  1.390700891 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.510426469
## 470  1.099985680 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.813578764
## 471  0.085480740 -0.48724019  1.01499462 -0.2723291  0.21837632 -0.167423167
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## 477  0.146239592 -0.48724019  1.01499462 -0.2723291  0.51178918  0.283747151
## 478  1.326491497 -0.48724019  1.01499462 -0.2723291  0.51178918 -1.395688102
## 479  0.769568300 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.141804663
## 480  1.246308229 -0.48724019  1.01499462 -0.2723291  0.51178918 -0.079181654
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## 482  0.243520951 -0.48724019  1.01499462 -0.2723291 -0.19585359  0.662331708
## 483  0.246192564 -0.48724019  1.01499462 -0.2723291 -0.19585359  1.104962525
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## 486  0.006992516 -0.48724019  1.01499462 -0.2723291  0.24426569  0.038948114
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## 488  0.142084524 -0.48724019  1.01499462 -0.2723291  0.24426569 -0.540314723
## 489 -0.402562972 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.182200570
## 490 -0.398783418 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.239130578
## 491 -0.395982758 -0.48724019  2.42017014 -0.2723291  0.46864023 -1.695993897
## 492 -0.407808541 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 493 -0.407159820 -0.48724019  2.42017014 -0.2723291  0.46864023 -0.429301206
## 494 -0.399952975 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.822118266
## 495 -0.387599381 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.510426469
## 496 -0.399292629 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.874778524
## 497 -0.386433311 -0.48724019 -0.21088983 -0.2723291  0.26152527 -1.273288584
## 498 -0.388900310 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.698295497
## 499 -0.392302024 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.378064199
## 500 -0.399427488 -0.48724019 -0.21088983 -0.2723291  0.26152527 -1.018526795
## 501 -0.394015670 -0.48724019 -0.21088983 -0.2723291  0.26152527 -0.366678197
## 502 -0.412820431 -0.48724019  0.11562398 -0.2723291  0.15796779  0.438881424
## 503 -0.414838673 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.234315927
## 504 -0.413037834 -0.48724019  0.11562398 -0.2723291  0.15796779  0.983986256
## 505 -0.407360947 -0.48724019  0.11562398 -0.2723291  0.15796779  0.724954717
## 506 -0.414589880 -0.48724019  0.11562398 -0.2723291  0.15796779 -0.362408446
##              age           dis        rad         tax     ptratio        black
## 1   -0.119894767  0.1400749840 -0.9818712 -0.66594918 -1.45755797  0.440615895
## 2    0.366803426  0.5566090496 -0.8670245 -0.98635338 -0.30279450  0.440615895
## 3   -0.265548971  0.5566090496 -0.8670245 -0.98635338 -0.30279450  0.396035074
## 4   -0.809087830  1.0766711351 -0.7521778 -1.10502160  0.11292035  0.415751408
## 5   -0.510674339  1.0766711351 -0.7521778 -1.10502160  0.11292035  0.440615895
## 6   -0.350809969  1.0766711351 -0.7521778 -1.10502160  0.11292035  0.410165113
## 7   -0.070159185  0.8384142195 -0.5224844 -0.57694801 -1.50374851  0.426376321
## 8    0.977840575  1.0236248974 -0.5224844 -0.57694801 -1.50374851  0.440615895
## 9    1.116389695  1.0861216287 -0.5224844 -0.57694801 -1.50374851  0.328123258
## 10   0.615481336  1.3283202075 -0.5224844 -0.57694801 -1.50374851  0.328999540
## 11   0.913894827  1.2117799501 -0.5224844 -0.57694801 -1.50374851  0.392639483
## 12   0.508905089  1.1547920492 -0.5224844 -0.57694801 -1.50374851  0.440615895
## 13  -1.050660656  0.7863652700 -0.5224844 -0.57694801 -1.50374851  0.370513376
## 14  -0.240681180  0.4333252240 -0.6373311 -0.60068166  1.17530274  0.440615895
## 15   0.565745754  0.3166899868 -0.6373311 -0.60068166  1.17530274  0.255720500
## 16  -0.428965883  0.3341187865 -0.6373311 -0.60068166  1.17530274  0.426595391
## 17  -1.395257187  0.3341187865 -0.6373311 -0.60068166  1.17530274  0.330533033
## 18   0.466274590  0.2198105553 -0.6373311 -0.60068166  1.17530274  0.329437681
## 19  -1.135921653  0.0006920764 -0.6373311 -0.60068166  1.17530274 -0.741378303
## 20   0.032864520  0.0006920764 -0.6373311 -0.60068166  1.17530274  0.375442459
## 21   1.048891406  0.0013569352 -0.6373311 -0.60068166  1.17530274  0.217930861
## 22   0.732715207  0.1031753182 -0.6373311 -0.60068166  1.17530274  0.392749018
## 23   0.821528746  0.0863638874 -0.6373311 -0.60068166  1.17530274  0.440615895
## 24   1.116389695  0.1425444597 -0.6373311 -0.60068166  1.17530274  0.414765591
## 25   0.906789743  0.2871037683 -0.6373311 -0.60068166  1.17530274  0.412465352
## 26   0.608376252  0.3132232229 -0.6373311 -0.60068166  1.17530274 -0.583319029
## 27   0.771793164  0.4212152950 -0.6373311 -0.60068166  1.17530274  0.221326452
## 28   0.718505041  0.3126533438 -0.6373311 -0.60068166  1.17530274 -0.550896614
## 29   0.917447368  0.3132707128 -0.6373311 -0.60068166  1.17530274  0.342472368
## 30   0.665216917  0.2108349609 -0.6373311 -0.60068166  1.17530274  0.258020739
## 31   0.906789743  0.2079855659 -0.6373311 -0.60068166  1.17530274  0.038293155
## 32   1.116389695  0.1804414138 -0.6373311 -0.60068166  1.17530274  0.219683424
## 33   0.476932215  0.0925850666 -0.6373311 -0.60068166  1.17530274 -1.359047220
## 34   0.938762618 -0.0037244859 -0.6373311 -0.60068166  1.17530274  0.022958229
## 35   1.006260907 -0.0167367233 -0.6373311 -0.60068166  1.17530274 -1.186967442
## 36  -0.013318520 -0.2064589434 -0.5224844 -0.76681717  0.34387304  0.440615895
## 37  -0.254891346 -0.1981007179 -0.5224844 -0.76681717  0.34387304  0.228774844
## 38  -0.961847117  0.0660856927 -0.5224844 -0.76681717  0.34387304  0.440615895
## 39  -1.363284313  0.0248169544 -0.5224844 -0.76681717  0.34387304  0.402607185
## 40  -1.661697804  0.7627152911 -0.7521778 -0.92701927 -0.07184181  0.426704926
## 41  -1.874850298  0.7627152911 -0.7521778 -0.92701927 -0.07184181  0.426595391
## 42  -2.333128159  0.9145880470 -0.7521778 -1.03975408 -0.25660396  0.314759966
## 43  -2.201684121  0.9145880470 -0.7521778 -1.03975408 -0.25660396  0.292414788
## 44  -2.205236663  0.9145880470 -0.7521778 -1.03975408 -0.25660396  0.413889309
## 45  -1.015135240  0.9145880470 -0.7521778 -1.03975408 -0.25660396  0.358354970
## 46  -1.235392817  0.6199131095 -0.7521778 -1.03975408 -0.25660396  0.440615895
## 47  -1.253155525  0.6199131095 -0.7521778 -1.03975408 -0.25660396  0.440615895
## 48   0.601271169  0.8996287230 -0.7521778 -1.03975408 -0.25660396  0.395049257
## 49   0.949420242  0.9853955139 -0.7521778 -1.03975408 -0.25660396  0.440615895
## 50  -0.233576097  1.0887810641 -0.7521778 -1.03975408 -0.25660396  0.440615895
## 51  -0.812640371  1.4340327636 -0.6373311 -0.98041997 -0.76469989  0.425938180
## 52  -0.198050682  1.4340327636 -0.6373311 -0.98041997 -0.76469989  0.408522085
## 53  -1.686565595  1.4340327636 -0.6373311 -0.98041997 -0.76469989  0.440615895
## 54  -1.675907970  1.4340327636 -0.6373311 -0.98041997 -0.76469989  0.440615895
## 55  -0.745142082  1.6738568465 -0.7521778  0.36053095  1.22149328  0.440615895
## 56  -1.658145262  2.3277455193 -0.5224844 -1.08128796 -0.25660396  0.429990982
## 57  -1.167894527  2.5609210138 -0.8670245 -0.56508119 -0.53374719  0.440615895
## 58  -0.997372532  2.1511780064 -0.5224844 -0.90328562 -1.54993904  0.396801820
## 59  -1.398809729  1.9089794276 -0.1779443 -0.73715011  0.57482574  0.372485009
## 60  -0.759352248  1.4897384367 -0.1779443 -0.73715011  0.57482574  0.440615895
## 61  -0.084369352  1.6290738544 -0.1779443 -0.73715011  0.57482574  0.421009097
## 62   0.881921953  1.4358373805 -0.1779443 -0.73715011  0.57482574  0.234470674
## 63  -0.027528687  1.6291213443 -0.1779443 -0.73715011  0.57482574  0.440615895
## 64  -0.894348827  1.9878601804 -0.1779443 -0.73715011  0.57482574  0.426157250
## 65  -0.322389636  2.5776849546 -0.7521778 -1.14062207  0.06672981  0.400526017
## 66  -1.803799466  1.3375332514 -0.6373311 -0.42267932 -1.08803366  0.440615895
## 67  -1.331311439  1.3375332514 -0.6373311 -0.42267932 -1.08803366  0.440615895
## 68  -1.675907970  1.2836321952 -0.6373311 -0.37521203  0.20530143  0.433057967
## 69  -1.128816570  1.2836321952 -0.6373311 -0.37521203  0.20530143  0.440615895
## 70  -1.263813149  1.2836321952 -0.6373311 -0.37521203  0.20530143  0.440615895
## 71  -2.201684121  0.7086717651 -0.6373311 -0.61254848  0.34387304  0.296358054
## 72  -1.814457091  0.7086717651 -0.6373311 -0.61254848  0.34387304  0.221983663
## 73  -2.159053622  0.7086717651 -0.6373311 -0.61254848  0.34387304  0.375004318
## 74  -2.215894287  0.7086717651 -0.6373311 -0.61254848  0.34387304  0.224502972
## 75  -2.222999370  0.2167712006 -0.5224844 -0.06074124  0.11292035  0.418927928
## 76  -0.837508162  0.3360183832 -0.5224844 -0.06074124  0.11292035  0.290881295
## 77   0.210491598  0.1221237952 -0.5224844 -0.06074124  0.11292035  0.186056121
## 78  -0.809087830  0.1403124336 -0.5224844 -0.06074124  0.11292035  0.331737920
## 79  -0.528437047  0.5789293108 -0.5224844 -0.06074124  0.11292035  0.325603949
## 80  -1.135921653  0.3360183832 -0.5224844 -0.06074124  0.11292035  0.431414939
## 81  -1.246050442  0.7625253315 -0.6373311 -0.75495035  0.25149196  0.440615895
## 82   0.064837394  0.7625253315 -0.6373311 -0.75495035  0.25149196  0.426704926
## 83  -1.292233482  0.7625253315 -0.6373311 -0.75495035  0.25149196  0.440615895
## 84  -0.777114956  0.7625253315 -0.6373311 -0.75495035  0.25149196  0.372046868
## 85  -0.730931915  0.4674704746 -0.7521778 -0.95668632  0.02053927  0.440615895
## 86  -0.443176049  0.3051974268 -0.7521778 -0.95668632  0.02053927  0.390229709
## 87  -0.833955621  0.3002109855 -0.7521778 -0.95668632  0.02053927  0.430648193
## 88  -0.418308258 -0.0225304932 -0.7521778 -0.95668632  0.02053927  0.421447237
## 89   0.629691502 -0.1773001341 -0.8670245 -0.82021787 -0.30279450  0.440615895
## 90  -0.194498140 -0.1807194081 -0.8670245 -0.82021787 -0.30279450  0.431414939
## 91  -0.087921893 -0.3337319220 -0.8670245 -0.82021787 -0.30279450  0.388915287
## 92   0.189176348 -0.3338269018 -0.8670245 -0.82021787 -0.30279450  0.403921608
## 93  -0.531989588 -0.0613297558 -0.6373311 -0.82021787 -0.11803234  0.419913745
## 94  -1.409467353 -0.0613297558 -0.6373311 -0.82021787 -0.11803234  0.434372389
## 95   0.309962761 -0.0855021237 -0.6373311 -0.82021787 -0.11803234  0.440615895
## 96  -0.382782843 -0.1423950448 -0.8670245 -0.78461740 -0.21041342  0.014304949
## 97   0.036417061 -0.1423950448 -0.8670245 -0.78461740 -0.21041342  0.385081555
## 98   0.263779721 -0.1423950448 -0.8670245 -0.78461740 -0.21041342  0.440615895
## 99  -1.125264029 -0.1423950448 -0.8670245 -0.78461740 -0.21041342  0.403702537
## 100 -0.215813389 -0.1423950448 -0.8670245 -0.78461740 -0.21041342  0.440615895
## 101  0.402328842 -0.4830877123 -0.5224844 -0.14380900  1.12911220  0.417175365
## 102  0.096810268 -0.4459031069 -0.5224844 -0.14380900  1.12911220  0.426157250
## 103  0.597718628 -0.5130538501 -0.5224844 -0.14380900  1.12911220 -3.131326538
## 104  0.668769459 -0.5130538501 -0.5224844 -0.14380900  1.12911220  0.413998845
## 105  0.761135540 -0.6525317376 -0.5224844 -0.14380900  1.12911220  0.394501581
## 106  0.999155824 -0.8016975682 -0.5224844 -0.14380900  1.12911220  0.409398367
## 107  0.828633829 -0.7522605641 -0.5224844 -0.14380900  1.12911220  0.427143067
## 108  0.590613545 -0.7943366310 -0.5224844 -0.14380900  1.12911220  0.339733988
## 109  1.013365990 -0.6468804374 -0.5224844 -0.14380900  1.12911220  0.422433054
## 110  0.803766038 -0.5935967501 -0.5224844 -0.14380900  1.12911220  0.378509444
## 111 -0.503569256 -0.4830877123 -0.5224844 -0.14380900  1.12911220  0.403264396
## 112  0.462722049 -0.5307200994 -0.4076377  0.14099473 -0.30279450  0.426266786
## 113  0.864159245 -0.6846349217 -0.4076377  0.14099473 -0.30279450  0.419256534
## 114  0.952972784 -0.5922195425 -0.4076377  0.14099473 -0.30279450  0.440615895
## 115  0.555088129 -0.7306526517 -0.4076377  0.14099473 -0.30279450  0.351235183
## 116  0.697189791 -0.6325384823 -0.4076377  0.14099473 -0.30279450 -0.128857540
## 117  0.139440767 -0.5057404029 -0.4076377  0.14099473 -0.30279450  0.401183228
## 118  0.498247464 -0.4975246471 -0.4076377  0.14099473 -0.30279450  0.414436985
## 119  0.160756016 -0.6256999342 -0.4076377  0.14099473 -0.30279450 -0.197645637
## 120 -0.119894767 -0.4919208369 -0.4076377  0.14099473 -0.30279450  0.381466894
## 121  0.039969603 -0.7300827727 -0.8670245 -1.30675758  0.29768250  0.355726125
## 122  0.551535588 -0.7587191929 -0.8670245 -1.30675758  0.29768250  0.229979731
## 123  0.864159245 -0.8111955516 -0.8670245 -1.30675758  0.29768250  0.234580209
## 124  1.009813449 -0.8788686839 -0.8670245 -1.30675758  0.29768250  0.149361834
## 125  0.967182950 -0.8494724251 -0.8670245 -1.30675758  0.29768250  0.248710248
## 126  0.704294875 -0.8558360740 -0.8670245 -1.30675758  0.29768250  0.310488093
## 127  0.960077867 -0.9677698093 -0.8670245 -1.30675758  0.29768250  0.028654058
## 128  0.974288033 -0.9530004450 -0.6373311  0.17066179  1.26768382  0.388148541
## 129  1.073759197 -0.9415078850 -0.6373311  0.17066179  1.26768382  0.440615895
## 130  0.928104993 -0.8620097633 -0.6373311  0.17066179  1.26768382  0.440615895
## 131  1.077311738 -0.7961887377 -0.6373311  0.17066179  1.26768382  0.420242350
## 132  1.034681240 -0.7237666137 -0.6373311  0.17066179  1.26768382  0.440615895
## 133  1.041786323 -0.6969823003 -0.6373311  0.17066179  1.26768382  0.318593697
## 134  0.952972784 -0.6293091680 -0.6373311  0.17066179  1.26768382  0.350687507
## 135  1.059549031 -0.6881491756 -0.6373311  0.17066179  1.26768382 -1.028689097
## 136  1.052443947 -0.7998929513 -0.6373311  0.17066179  1.26768382  0.416189548
## 137  0.885474494 -0.8681834525 -0.6373311  0.17066179  1.26768382  0.236332772
## 138  1.059549031 -0.9237941458 -0.6373311  0.17066179  1.26768382  0.409726972
## 139  1.052443947 -1.0098458762 -0.6373311  0.17066179  1.26768382  0.387381794
## 140  1.041786323 -1.0097983862 -0.6373311  0.17066179  1.26768382  0.440615895
## 141  0.889027036 -1.0367726593 -0.6373311  0.17066179  1.26768382  0.344005860
## 142  1.116389695 -1.1186927669 -0.6373311  0.17066179  1.26768382  0.440615895
## 143  1.116389695 -1.1746358896 -0.5224844 -0.03107419 -1.73470120  0.440615895
## 144  1.116389695 -1.1317999841 -0.5224844 -0.03107419 -1.73470120  0.440615895
## 145  1.038233781 -1.1630958396 -0.5224844 -0.03107419 -1.73470120  0.440615895
## 146  1.116389695 -1.1283332201 -0.5224844 -0.03107419 -1.73470120 -2.012862748
## 147  1.116389695 -1.0820305506 -0.5224844 -0.03107419 -1.73470120 -2.052733556
## 148  0.963630408 -1.1085299245 -0.5224844 -0.03107419 -1.73470120  0.383767133
## 149  0.896132119 -1.0758568614 -0.5224844 -0.03107419 -1.73470120  0.003460966
## 150  0.935210076 -1.0777089681 -0.5224844 -0.03107419 -1.73470120 -0.052840120
## 151  1.020471073 -1.0338757744 -0.5224844 -0.03107419 -1.73470120  0.176636095
## 152  1.116389695 -1.0464131126 -0.5224844 -0.03107419 -1.73470120 -0.165113687
## 153  0.690084708 -1.0375799879 -0.5224844 -0.03107419 -1.73470120 -0.146711775
## 154  1.063101572 -1.0314062987 -0.5224844 -0.03107419 -1.73470120 -1.037561447
## 155  0.974288033 -0.9714740229 -0.5224844 -0.03107419 -1.73470120 -0.390537100
## 156  0.498247464 -0.9733261297 -0.5224844 -0.03107419 -1.73470120 -2.942816482
## 157  0.903237202 -0.9776477121 -0.5224844 -0.03107419 -1.73470120 -2.936025300
## 158  1.024023615 -0.9107344185 -0.5224844 -0.03107419 -1.73470120  0.074001626
## 159  1.116389695 -0.9677223194 -0.5224844 -0.03107419 -1.73470120 -0.030494942
## 160  1.116389695 -0.9636381865 -0.5224844 -0.03107419 -1.73470120  0.083640722
## 161  0.853501620 -0.9482039634 -0.5224844 -0.03107419 -1.73470120 -0.194469117
## 162  0.789555872 -0.8662838558 -0.5224844 -0.03107419 -1.73470120  0.194490331
## 163  1.052443947 -0.8331358935 -0.5224844 -0.03107419 -1.73470120  0.360764744
## 164  0.899684660 -0.7755306237 -0.5224844 -0.03107419 -1.73470120  0.348058662
## 165  0.825081288 -0.6520568384 -0.5224844 -0.03107419 -1.73470120  0.421009097
## 166  0.867711786 -0.7178778639 -0.5224844 -0.03107419 -1.73470120 -1.276238619
## 167  0.981393116 -0.8306664178 -0.5224844 -0.03107419 -1.73470120  0.138298780
## 168  0.377461051 -0.6502047316 -0.5224844 -0.03107419 -1.73470120 -1.413705278
## 169  0.977840575 -0.8049743725 -0.5224844 -0.03107419 -1.73470120 -0.652654802
## 170  0.945867701 -0.7278032567 -0.5224844 -0.03107419 -1.73470120 -0.291736362
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## 353 -1.778931675  3.2840499862 -0.6373311  0.01639310 -0.07184181  0.390558315
## 354 -1.153684361  3.9566021965 -0.5224844 -1.31269099 -0.67231881  0.304354123
## 355 -1.658145262  3.2248775491 -0.6373311 -0.44047956  1.63720813  0.286171282
## 356 -1.743406260  3.2248775491 -0.6373311 -0.44047956  1.63720813  0.212125496
## 357  1.024023615 -0.7944316108  1.6596029  1.52941294  0.80577843  0.230636942
## 358  0.796660955 -0.6125452271  1.6596029  1.52941294  0.80577843  0.379714331
## 359  0.526667797 -0.5092546567  1.6596029  1.52941294  0.80577843  0.424514223
## 360  0.452064424 -0.6106931203  1.6596029  1.52941294  0.80577843  0.373142220
## 361  0.690084708 -0.6063715378  1.6596029  1.52941294  0.80577843  0.195914288
## 362  0.800213497 -0.7121315839  1.6596029  1.52941294  0.80577843 -0.065984343
## 363  0.981393116 -0.8032647354  1.6596029  1.52941294  0.80577843  0.264154709
## 364  0.725610124 -0.8977221812  1.6596029  1.52941294  0.80577843 -0.039805433
## 365  0.508905089 -0.8977221812  1.6596029  1.52941294  0.80577843 -0.023265620
## 366  0.686532167 -1.0361552904  1.6596029  1.52941294  0.80577843 -0.021622592
## 367  0.810871121 -0.9700968153  1.6596029  1.52941294  0.80577843 -0.445195159
## 368  1.116389695 -1.0848799457  1.6596029  1.52941294  0.80577843 -2.467324237
## 369  1.116389695 -1.1694594886  1.6596029  1.52941294  0.80577843  0.206429666
## 370  1.002708366 -1.1579669285  1.6596029  1.52941294  0.80577843  0.204348498
## 371  1.027576156 -1.2312438711  1.6596029  1.52941294  0.80577843  0.387491330
## 372  1.116389695 -1.2470580136  1.6596029  1.52941294  0.80577843  0.103795196
## 373  0.746925373 -1.2658165310  1.6596029  1.52941294  0.80577843 -0.096325589
## 374  1.116389695 -1.2446360278  1.6596029  1.52941294  0.80577843  0.440615895
## 375  1.116389695 -1.2623022771  1.6596029  1.52941294  0.80577843  0.440615895
## 376  1.041786323 -1.1771528552  1.6596029  1.52941294  0.80577843  0.440615895
## 377  0.878369411 -1.1635707388  1.6596029  1.52941294  0.80577843  0.069510683
## 378  1.073759197 -1.1573495596  1.6596029  1.52941294  0.80577843  0.440615895
## 379  0.981393116 -1.1440048928  1.6596029  1.52941294  0.80577843  0.440615895
## 380  1.116389695 -1.1440048928  1.6596029  1.52941294  0.80577843  0.406002776
## 381  0.828633829 -1.1295679579  1.6596029  1.52941294  0.80577843  0.440615895
## 382  1.084416821 -1.0807958128  1.6596029  1.52941294  0.80577843  0.440615895
## 383  1.116389695 -1.0517319833  1.6596029  1.52941294  0.80577843  0.440615895
## 384  1.116389695 -1.0741947142  1.6596029  1.52941294  0.80577843  0.440615895
## 385  0.803766038 -1.1186452769  1.6596029  1.52941294  0.80577843 -0.775991422
## 386  1.048891406 -1.1250089259  1.6596029  1.52941294  0.80577843  0.440615895
## 387  1.116389695 -1.1054905698  1.6596029  1.52941294  0.80577843  0.440615895
## 388  0.743372832 -1.0811757321  1.6596029  1.52941294  0.80577843  0.440615895
## 389  1.116389695 -1.0474104008  1.6596029  1.52941294  0.80577843  0.177950518
## 390  1.077311738 -0.9815893753  1.6596029  1.52941294  0.80577843  0.440615895
## 391  1.009813449 -0.8873693792  1.6596029  1.52941294  0.80577843  0.413560704
## 392  0.494694923 -0.7727762084  1.6596029  1.52941294  0.80577843  0.237756730
## 393  1.009813449 -0.9616910999  1.6596029  1.52941294  0.80577843  0.440615895
## 394  0.853501620 -0.9516232374  1.6596029  1.52941294  0.80577843  0.440615895
## 395  0.928104993 -0.9559448199  1.6596029  1.52941294  0.80577843  0.440615895
## 396  1.073759197 -0.9827291333  1.6596029  1.52941294  0.80577843  0.386724583
## 397  0.974288033 -1.0059517029  1.6596029  1.52941294  0.80577843  0.440615895
## 398  1.077311738 -1.0265623271  1.6596029  1.52941294  0.80577843  0.398992524
## 399  1.116389695 -1.0948528283  1.6596029  1.52941294  0.80577843  0.440615895
## 400  0.327725469 -1.0897239172  1.6596029  1.52941294  0.80577843 -0.202793791
## 401  1.116389695 -1.0477428302  1.6596029  1.52941294  0.80577843  0.440615895
## 402  1.116389695 -1.0547238481  1.6596029  1.52941294  0.80577843  0.440615895
## 403  1.116389695 -1.0239028917  1.6596029  1.52941294  0.80577843  0.212892242
## 404  0.974288033 -0.9936043244  1.6596029  1.52941294  0.80577843  0.440615895
## 405  0.597718628 -1.0389097056  1.6596029  1.52941294  0.80577843 -0.298089403
## 406  1.116389695 -1.1253413553  1.6596029  1.52941294  0.80577843  0.309940417
## 407  1.116389695 -1.2427839210  1.6596029  1.52941294  0.80577843  0.148376017
## 408  1.116389695 -1.1919222195  1.6596029  1.52941294  0.80577843 -0.269281649
## 409  1.041786323 -1.1114268095  1.6596029  1.52941294  0.80577843 -0.460420549
## 410  1.116389695 -1.1062978984  1.6596029  1.52941294  0.80577843 -1.942212553
## 411  1.116389695 -1.1312301050  1.6596029  1.52941294  0.80577843 -3.878356510
## 412  1.116389695 -1.0768541496  1.6596029  1.52941294  0.80577843 -3.522914830
## 413  1.116389695 -1.0643168114  1.6596029  1.52941294  0.80577843 -3.591483857
## 414  1.116389695 -1.0474578907  1.6596029  1.52941294  0.80577843 -1.595971828
## 415  1.116389695 -1.0147848276  1.6596029  1.52941294  0.80577843 -2.939968567
## 416  1.116389695 -0.9309651233  1.6596029  1.52941294  0.80577843 -3.608352275
## 417  0.789555872 -0.9381835908  1.6596029  1.52941294  0.80577843 -3.670568261
## 418  0.729162665 -1.0198662487  1.6596029  1.52941294  0.80577843 -2.511795523
## 419  1.116389695 -0.9462093868  1.6596029  1.52941294  0.80577843 -3.726650277
## 420  0.281542429 -0.9502935197  1.6596029  1.52941294  0.80577843 -3.376137680
## 421  1.116389695 -0.9194725633  1.6596029  1.52941294  0.80577843 -0.415401588
## 422  0.949420242 -0.9120166463  1.6596029  1.52941294  0.80577843 -0.401928760
## 423  0.675874542 -0.8756393696  1.6596029  1.52941294  0.80577843 -0.713337295
## 424  0.587061003 -0.8421114879  1.6596029  1.52941294  0.80577843 -3.879232792
## 425  0.071942477 -0.8223081923  1.6596029  1.52941294  0.80577843 -3.866855315
## 426  0.952972784 -0.8953951752  1.6596029  1.52941294  0.80577843 -3.822712635
## 427 -0.315284553 -0.8536040479  1.6596029  1.52941294  0.80577843 -3.636831424
## 428  0.359698343 -0.9175729666  1.6596029  1.52941294  0.80577843 -3.700690438
## 429  0.338383094 -0.8830477967  1.6596029  1.52941294  0.80577843 -2.847301799
## 430  0.960077867 -0.8675660836  1.6596029  1.52941294  0.80577843 -3.241738006
## 431  0.622586419 -0.8274371034  1.6596029  1.52941294  0.80577843 -2.992764527
## 432  0.913894827 -0.8105781827  1.6596029  1.52941294  0.80577843 -3.015985986
## 433  0.221149222 -0.7572944954  1.6596029  1.52941294  0.80577843 -2.833938506
## 434  0.686532167 -0.7024911307  1.6596029  1.52941294  0.80577843 -2.809402625
## 435  0.938762618 -0.7469416934  1.6596029  1.52941294  0.80577843 -2.804583076
## 436  0.924552451 -0.7932443629  1.6596029  1.52941294  0.80577843 -2.703591634
## 437  0.878369411 -0.8512295520  1.6596029  1.52941294  0.80577843 -3.605723431
## 438  1.116389695 -0.8932106390  1.6596029  1.52941294  0.80577843 -3.804748865
## 439  0.686532167 -0.9376612017  1.6596029  1.52941294  0.80577843 -3.151590547
## 440  0.899684660 -0.9392758589  1.6596029  1.52941294  0.80577843  0.440615895
## 441  0.846396537 -0.9160057994  1.6596029  1.52941294  0.80577843  0.380919218
## 442  1.016918532 -0.8215483536  1.6596029  1.52941294  0.80577843  0.320784401
## 443  1.116389695 -0.8501847738  1.6596029  1.52941294  0.80577843  0.427362137
## 444  1.116389695 -0.8627221120  1.6596029  1.52941294  0.80577843  0.329218610
## 445  0.995603282 -0.9020437636  1.6596029  1.52941294  0.80577843 -1.272295352
## 446  0.931657534 -0.8582105699  1.6596029  1.52941294  0.80577843 -3.435177145
## 447  0.988498199 -0.8182715493  1.6596029  1.52941294  0.80577843 -0.423507192
## 448  0.995603282 -0.7584342534  1.6596029  1.52941294  0.80577843  0.348825409
## 449  1.070206655 -0.7282306659  1.6596029  1.52941294  0.80577843  0.440615895
## 450  1.055996489 -0.7646079427  1.6596029  1.52941294  0.80577843 -0.574665749
## 451  0.853501620 -0.6987869171  1.6596029  1.52941294  0.80577843 -3.903330533
## 452  1.052443947 -0.6837801032  1.6596029  1.52941294  0.80577843 -0.015160016
## 453  0.825081288 -0.6776064140  1.6596029  1.52941294  0.80577843  0.311254840
## 454  1.091521905 -0.6374774338  1.6596029  1.52941294  0.80577843  0.210263398
## 455  0.906789743 -0.6168668096  1.6596029  1.52941294  0.80577843 -3.833666154
## 456  0.636796585 -0.6455032298  1.6596029  1.52941294  0.80577843 -3.349082489
## 457  0.686532167 -0.5767378294  1.6596029  1.52941294  0.80577843 -3.792042783
## 458  0.416539008 -0.4824228534  1.6596029  1.52941294  0.80577843 -3.868498343
## 459  0.537325421 -0.4805707466  1.6596029  1.52941294  0.80577843 -0.925178346
## 460  0.562193212 -0.5117241325  1.6596029  1.52941294  0.80577843  0.440615895
## 461  0.761135540 -0.5687120333  1.6596029  1.52941294  0.80577843 -1.111169093
## 462  0.704294875 -0.5831489682  1.6596029  1.52941294  0.80577843  0.380700148
## 463  0.512457630 -0.5036983364  1.6596029  1.52941294  0.80577843  0.440615895
## 464  0.757582998 -0.4717851119  1.6596029  1.52941294  0.80577843  0.406879058
## 465 -0.112789684 -0.3949464255  1.6596029  1.52941294  0.80577843  0.440615895
## 466 -0.723826832 -0.3459843207  1.6596029  1.52941294  0.80577843 -0.243979021
## 467  0.572850837 -0.4385896596  1.6596029  1.52941294  0.80577843 -3.665748713
## 468  0.920999910 -0.5958762661  1.6596029  1.52941294  0.80577843 -0.278044464
## 469  0.086152643 -0.4210658801  1.6596029  1.52941294  0.80577843  0.132164810
## 470 -0.421860800 -0.4612898402  1.6596029  1.52941294  0.80577843  0.440615895
## 471  0.547983046 -0.3617034834  1.6596029  1.52941294  0.80577843  0.440615895
## 472  0.786003330 -0.3304076278  1.6596029  1.52941294  0.80577843  0.423418871
## 473  0.228254306 -0.4267171803  1.6596029  1.52941294  0.80577843  0.401949974
## 474 -0.034633770 -0.5993905200  1.6596029  1.52941294  0.80577843  0.197228711
## 475  0.952972784 -0.6483526248  1.6596029  1.52941294  0.80577843 -0.044844052
## 476  1.024023615 -0.7546350600  1.6596029  1.52941294  0.80577843 -0.590548351
## 477  0.889027036 -0.7074775720  1.6596029  1.52941294  0.80577843  0.433057967
## 478  1.020471073 -0.8046419430  1.6596029  1.52941294  0.80577843 -0.078799960
## 479  0.999155824 -0.7714939807  1.6596029  1.52941294  0.80577843  0.252215374
## 480  0.690084708 -0.8756393696  1.6596029  1.52941294  0.80577843  0.291867112
## 481 -0.137657475 -0.1761128861  1.6596029  1.52941294  0.80577843  0.440615895
## 482  0.224701764 -0.2200410597  1.6596029  1.52941294  0.80577843  0.398663919
## 483  0.299305137 -0.1825715149  1.6596029  1.52941294  0.80577843  0.422871195
## 484 -1.004477616  0.1440166471  1.6596029  1.52941294  0.80577843  0.397020891
## 485 -0.947636951 -0.0337381137  1.6596029  1.52941294  0.80577843  0.153962312
## 486 -0.592382795  0.0933923952  1.6596029  1.52941294  0.80577843  0.349920761
## 487  0.398776300 -0.1183176566  1.6596029  1.52941294  0.80577843  0.394392046
## 488 -0.546199754 -0.3052379716  1.6596029  1.52941294  0.80577843  0.345539353
## 489  0.857054162 -0.9375187319 -0.6373311  1.79641644  0.75958789  0.420790026
## 490  1.055996489 -0.9686246278 -0.6373311  1.79641644  0.75958789 -0.138277566
## 491  1.045338864 -0.9367114033 -0.6373311  1.79641644  0.75958789 -0.418906714
## 492  1.073759197 -0.9151034909 -0.6373311  1.79641644  0.75958789  0.366241503
## 493  0.530220338 -0.8002728706 -0.6373311  1.79641644  0.75958789  0.440615895
## 494 -0.517779422 -0.6711952751 -0.4076377 -0.10227512  0.34387304  0.440615895
## 495 -0.922769160 -0.6711952751 -0.4076377 -0.10227512  0.34387304  0.440615895
## 496 -1.413019895 -0.4732098094 -0.4076377 -0.10227512  0.34387304  0.401073693
## 497  0.153650933 -0.4732098094 -0.4076377 -0.10227512  0.34387304  0.440615895
## 498  0.071942477 -0.4285217972 -0.4076377 -0.10227512  0.34387304  0.440615895
## 499 -0.116342226 -0.6581830377 -0.4076377 -0.10227512  0.34387304  0.440615895
## 500  0.174966182 -0.6625521101 -0.4076377 -0.10227512  0.34387304  0.428238419
## 501  0.395223759 -0.6158695213 -0.4076377 -0.10227512  0.34387304  0.440615895
## 502  0.018654354 -0.6251775451 -0.9818712 -0.80241764  1.17530274  0.386834118
## 503  0.288647512 -0.7159307773 -0.9818712 -0.80241764  1.17530274  0.440615895
## 504  0.796660955 -0.7729186782 -0.9818712 -0.80241764  1.17530274  0.440615895
## 505  0.736267749 -0.6677760011 -0.9818712 -0.80241764  1.17530274  0.402826256
## 506  0.434301716 -0.6126402069 -0.9818712 -0.80241764  1.17530274  0.440615895
##            lstat         medv
## 1   -1.074498970  0.159527789
## 2   -0.491952525 -0.101423917
## 3   -1.207532413  1.322937477
## 4   -1.360170785  1.181588636
## 5   -1.025486649  1.486032293
## 6   -1.042290874  0.670558212
## 7   -0.031236706  0.039924923
## 8    0.909799859  0.496590409
## 9    2.419379350 -0.655946292
## 10   0.622727693 -0.394994586
## 11   1.091845624 -0.819041108
## 12   0.086392864 -0.394994586
## 13   0.428078760 -0.090550930
## 14  -0.615183504 -0.231899770
## 15  -0.335113097 -0.471105500
## 16  -0.585776111 -0.286264709
## 17  -0.850442645  0.061670899
## 18   0.282442149 -0.547216415
## 19  -0.134862757 -0.253645746
## 20  -0.192277190 -0.471105500
## 21   1.171665689 -0.971262937
## 22   0.164812578 -0.318883672
## 23   0.849584722 -0.797295133
## 24   1.012025558 -0.873406047
## 25   0.510699530 -0.753803182
## 26   0.540106923 -0.938643973
## 27   0.302047077 -0.645073304
## 28   0.647934029 -0.840787084
## 29   0.020576319 -0.449359525
## 30  -0.094252548 -0.166661844
## 31   1.392921311 -1.069119826
## 32   0.054184768 -0.873406047
## 33   2.108501199 -1.014754888
## 34   0.797771697 -1.025627875
## 35   1.076441751 -0.982135924
## 36  -0.416333515 -0.394994586
## 37  -0.174072614 -0.275391721
## 38  -0.543765550 -0.166661844
## 39  -0.353317674  0.235638703
## 40  -1.166922204  0.898890955
## 41  -1.494604580  1.344683452
## 42  -1.094103899  0.442225470
## 43  -0.958269752  0.300876629
## 44  -0.730012370  0.235638703
## 45  -0.434538092 -0.144915868
## 46  -0.342114857 -0.351502635
## 47   0.209623843 -0.275391721
## 48   0.860787538 -0.645073304
## 49   2.542610329 -0.884279035
## 50   0.496696010 -0.340629648
## 51   0.111599201 -0.308010684
## 52  -0.451342316 -0.221026782
## 53  -1.032488409  0.268257666
## 54  -0.591377519  0.094289862
## 55   0.300646725 -0.394994586
## 56  -1.098304955  1.399048391
## 57  -0.963871160  0.235638703
## 58  -1.218735230  0.985874857
## 59  -0.811232788  0.083416874
## 60  -0.480749709 -0.318883672
## 61   0.069588640 -0.416740562
## 62   0.250234052 -0.710311231
## 63  -0.829437365 -0.036185991
## 64  -0.441539852  0.268257666
## 65  -0.644590896  1.138096685
## 66  -1.117909883  0.105162850
## 67  -0.337913801 -0.340629648
## 68  -0.637589136 -0.057931966
## 69   0.061186528 -0.558089402
## 70  -0.540964846 -0.177534831
## 71  -0.830837717  0.181273764
## 72  -0.388326475 -0.090550930
## 73  -0.998879961  0.029051936
## 74  -0.716008850  0.094289862
## 75  -0.822435605  0.170400776
## 76  -0.519959566 -0.123169893
## 77  -0.095652900 -0.275391721
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boston_scaled <- as.data.frame(scale(Boston))
boston_scaled$crim <- as.numeric(boston_scaled$crim)
# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
#Create a categorical variable of the crime rate
crime <- cut(boston_scaled$crim, 
             breaks = bins, 
             include.lowest = TRUE,
             label = c("low", "medium_low", "medium_high", "high"))
summary(crime)
##         low  medium_low medium_high        high 
##         127         126         126         127
# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)

# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)

Here begins the splitting of the data into a training set and a test set, with a split ratio of 80% of the training set data and 20% of the test set data.

# number of rows in the Boston dataset 
n <- nrow(boston_scaled)

# choose randomly 80% of the rows
ind <- sample(n,  size = n * 0.8)

# create train set
train <- boston_scaled[ind,]

# create test set 
test <- boston_scaled[-ind,]

# save the correct classes from test data
correct_classes <- boston_scaled$crime

# remove the crime variable from test data
test <- dplyr::select(test, -crime)

summary(train)
##        zn               indus               chas               nox          
##  Min.   :-0.48724   Min.   :-1.55630   Min.   :-0.27233   Min.   :-1.42991  
##  1st Qu.:-0.48724   1st Qu.:-0.87558   1st Qu.:-0.27233   1st Qu.:-0.92076  
##  Median :-0.48724   Median :-0.37560   Median :-0.27233   Median :-0.23037  
##  Mean   : 0.01009   Mean   :-0.03768   Mean   : 0.02003   Mean   :-0.03642  
##  3rd Qu.: 0.28991   3rd Qu.: 1.01499   3rd Qu.:-0.27233   3rd Qu.: 0.51179  
##  Max.   : 3.80047   Max.   : 2.42017   Max.   : 3.66477   Max.   : 2.72965  
##        rm                age                dis                rad          
##  Min.   :-3.44659   Min.   :-2.33313   Min.   :-1.26582   Min.   :-0.98187  
##  1st Qu.:-0.56451   1st Qu.:-0.95119   1st Qu.:-0.79695   1st Qu.:-0.63733  
##  Median :-0.05001   Median : 0.21582   Median :-0.25055   Median :-0.52248  
##  Mean   : 0.02716   Mean   :-0.03974   Mean   : 0.01816   Mean   :-0.05713  
##  3rd Qu.: 0.49439   3rd Qu.: 0.90146   3rd Qu.: 0.70867   3rd Qu.:-0.17794  
##  Max.   : 3.55153   Max.   : 1.11639   Max.   : 3.95660   Max.   : 1.65960  
##       tax             ptratio             black              lstat         
##  Min.   :-1.3127   Min.   :-2.70470   Min.   :-3.87923   Min.   :-1.52961  
##  1st Qu.:-0.7787   1st Qu.:-0.68387   1st Qu.: 0.21900   1st Qu.:-0.84169  
##  Median :-0.4701   Median : 0.18221   Median : 0.38327   Median :-0.24899  
##  Mean   :-0.0465   Mean   :-0.02759   Mean   : 0.03728   Mean   :-0.03607  
##  3rd Qu.: 0.2181   3rd Qu.: 0.80578   3rd Qu.: 0.43248   3rd Qu.: 0.55341  
##  Max.   : 1.7964   Max.   : 1.63721   Max.   : 0.44062   Max.   : 3.54526  
##       medv                  crime    
##  Min.   :-1.90634   low        :104  
##  1st Qu.:-0.55809   medium_low :107  
##  Median :-0.09599   medium_high:103  
##  Mean   : 0.03984   high       : 90  
##  3rd Qu.: 0.32534                    
##  Max.   : 2.98650
summary(test)
##        zn               indus              chas               nox         
##  Min.   :-0.48724   Min.   :-1.4018   Min.   :-0.27233   Min.   :-1.4644  
##  1st Qu.:-0.48724   1st Qu.:-0.7932   1st Qu.:-0.27233   1st Qu.:-0.7827  
##  Median :-0.48724   Median : 0.3263   Median :-0.27233   Median :-0.1182  
##  Mean   :-0.03997   Mean   : 0.1492   Mean   :-0.07933   Mean   : 0.1443  
##  3rd Qu.:-0.48724   3rd Qu.: 1.0150   3rd Qu.:-0.27233   3rd Qu.: 1.1633  
##  Max.   : 3.58609   Max.   : 2.1155   Max.   : 3.66477   Max.   : 2.7296  
##        rm               age               dis                rad         
##  Min.   :-3.8764   Min.   :-1.8749   Min.   :-1.24706   Min.   :-0.9819  
##  1st Qu.:-0.5766   1st Qu.:-0.5036   1st Qu.:-0.87223   1st Qu.:-0.6373  
##  Median :-0.2379   Median : 0.4787   Median :-0.44092   Median :-0.5225  
##  Mean   :-0.1076   Mean   : 0.1574   Mean   :-0.07194   Mean   : 0.2263  
##  3rd Qu.: 0.2443   3rd Qu.: 0.9139   3rd Qu.: 0.43181   3rd Qu.: 1.6596  
##  Max.   : 2.9751   Max.   : 1.1164   Max.   : 3.28405   Max.   : 1.6596  
##       tax             ptratio            black              lstat         
##  Min.   :-1.3068   Min.   :-2.7047   Min.   :-3.90333   Min.   :-1.49460  
##  1st Qu.:-0.7015   1st Qu.:-0.3028   1st Qu.: 0.09114   1st Qu.:-0.58893  
##  Median :-0.1023   Median : 0.5517   Median : 0.36756   Median : 0.05418  
##  Mean   : 0.1842   Mean   : 0.1093   Mean   :-0.14766   Mean   : 0.14286  
##  3rd Qu.: 1.5294   3rd Qu.: 0.8058   3rd Qu.: 0.44062   3rd Qu.: 0.72180  
##  Max.   : 1.5294   Max.   : 1.2677   Max.   : 0.44062   Max.   : 2.99212  
##       medv         
##  Min.   :-1.90634  
##  1st Qu.:-0.91962  
##  Median :-0.23734  
##  Mean   :-0.15781  
##  3rd Qu.: 0.06983  
##  Max.   : 2.98650
nrow(train)
## [1] 404
nrow(test)
## [1] 102

4.Fit the linear discriminant analysis on the train set. Use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables.

# linear discriminant analysis
lda.fit <- lda(crime ~., data = train)

# print the lda.fit object
lda.fit 
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##         low  medium_low medium_high        high 
##   0.2574257   0.2648515   0.2549505   0.2227723 
## 
## Group means:
##                      zn      indus         chas        nox          rm
## low          0.92667462 -0.9075068 -0.083045403 -0.8573768  0.48244109
## medium_low  -0.08952469 -0.3008146 -0.014761763 -0.5566191 -0.13490123
## medium_high -0.37734151  0.1920304  0.186362222  0.3810018  0.08592919
## high        -0.48724019  1.0173916 -0.009855719  1.0529729 -0.37352561
##                    age        dis        rad        tax     ptratio      black
## low         -0.8750490  0.8382480 -0.6947544 -0.7411438 -0.42182396  0.3875640
## medium_low  -0.3786327  0.3487745 -0.5396577 -0.4414222 -0.06018625  0.3234368
## medium_high  0.3628025 -0.3718571 -0.3909124 -0.2966951 -0.29920689  0.1063241
## high         0.8677118 -0.8761908  1.6353575  1.5120742  0.77755088 -0.7867161
##                    lstat        medv
## low         -0.804311818  0.55555546
## medium_low  -0.138631929 -0.01626912
## medium_high  0.009699798  0.18380728
## high         0.921236068 -0.65413413
## 
## Coefficients of linear discriminants:
##                  LD1          LD2        LD3
## zn       0.096839438  0.708480304 -0.9515761
## indus    0.003268679 -0.352029108  0.1835904
## chas    -0.073261998 -0.006963925  0.1228697
## nox      0.412626040 -0.730228820 -1.2780645
## rm      -0.071745585 -0.056208521 -0.1292698
## age      0.279320295 -0.214769866 -0.1626182
## dis     -0.057277936 -0.261442982  0.2002652
## rad      2.999099874  0.828856469 -0.2302545
## tax     -0.060794191  0.142282445  0.8078329
## ptratio  0.112870581  0.066923816 -0.2835724
## black   -0.165829572  0.015733646  0.1806925
## lstat    0.245910553 -0.275867823  0.3465201
## medv     0.195334063 -0.460373155 -0.2654702
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9429 0.0408 0.0162
  1. Save the crime categories from the test set and then remove the categorical crime variable from the test dataset. Then predict the classes with the LDA model on the test data.
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(train$crime)

# plot the lda results
plot(lda.fit, dimen = 2)
lda.arrows(lda.fit, myscale = 1)

In this graph, the clustering of crime levels can be clearly seen, with low and medium_low crime rates concentrated in the top left and middle left; medium_high and high crime rates are concentrated in the bottom left and right. The concentration of low crime is much greater, and there is a very clear demarcation line between high crime on the right and the left. In the red part of the graph, rad(index of accessibility to radial highway) is associated with high crime rates and zn(proportion of residential land zoned for lots over 25,000 sq.ft) is closely related to low crime rates.

# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)

# cross tabulate the results
predictions <- lda.pred$class
#table(correct = correct_classes, predicted = predictions)

6.Calculate the distances between the observations. Run k-means algorithm on the dataset.

library(MASS)
#reload the Boston dataset and standardize the detaset
data("Boston") 
boston_scaled <- as.data.frame(scale(Boston))

#calculate the distances between the observations
dist <- dist(boston_scaled)
summary(dist)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970

k-means

set.seed(13)
km <- kmeans(Boston, centers = 4)

# plot the Boston dataset with clusters
km <- kmeans(Boston, centers = 4)

# plot the Boston dataset with clusters
pairs(Boston[6:10], col = km$cluster)

pairs(Boston[1:5], col = km$cluster)

pairs(Boston[11:14], col = km$cluster)

set.seed(123)

# determine the number of clusters
k_max <- 10

# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})

# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')

# k-means clustering
km <- kmeans(Boston, centers = 2)

# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)

The optimal number of clusters can be found by looking at the sum of squares within clusters, and in the graph above I think the optimal number of clusters is close to 2. At an optimal cluster number of 2, the values vary significantly.


Prepare some package

library(dplyr)
library(tidyverse)
library(GGally)

Read data

human <- read.table("human.txt", sep = ",", )
dim(human)
## [1] 155   8

The dataset has 155 observations and 8 variables.

  1. Show a graphical overview of the data and show summaries of the variables in the data.
ggpairs(human)

summary(human)
##      edu_fm         labour_fm         edu.exp         life.exp    
##  Min.   :0.1717   Min.   :0.1857   Min.   : 5.40   Min.   :49.00  
##  1st Qu.:0.7264   1st Qu.:0.5984   1st Qu.:11.25   1st Qu.:66.30  
##  Median :0.9375   Median :0.7535   Median :13.50   Median :74.20  
##  Mean   :0.8529   Mean   :0.7074   Mean   :13.18   Mean   :71.65  
##  3rd Qu.:0.9968   3rd Qu.:0.8535   3rd Qu.:15.20   3rd Qu.:77.25  
##  Max.   :1.4967   Max.   :1.0380   Max.   :20.20   Max.   :83.50  
##       gni         maternal.mortality.ratio   teen.birth         per.pa     
##  Min.   :   581   Min.   :   1.0           Min.   :  0.60   Min.   : 0.00  
##  1st Qu.:  4198   1st Qu.:  11.5           1st Qu.: 12.65   1st Qu.:12.40  
##  Median : 12040   Median :  49.0           Median : 33.60   Median :19.30  
##  Mean   : 17628   Mean   : 149.1           Mean   : 47.16   Mean   :20.91  
##  3rd Qu.: 24512   3rd Qu.: 190.0           3rd Qu.: 71.95   3rd Qu.:27.95  
##  Max.   :123124   Max.   :1100.0           Max.   :204.80   Max.   :57.50

The correlation between the variables can be seen very clearly in the visual charts. Take the variable of the labour market participation of females and males (labour_fm) as an example, there is a strong correlation between it and the percentage of parliamentary representation (per.pa), maternal mortality rate (maternal.mortality.ratio). As can be seen from the scatter plot, there appears to be a positive correlation between labour_fm and per.pa and maternal.mortality.ratio.

# Access corrplot
library(corrplot)
# compute the correlation matrix and visualize it with corrplot
cor(human) %>%
  corrplot(method = "number")

In a heat map, the degree of correlation and correlations between variables can be seen more visually through the figures.

2-3Perform principal component analysis (PCA) on the raw (non-standardized) human data.

# standardize the variables
human_std <- scale(human)

# print out summaries of the standardized variables
summary(human_std)
##      edu_fm          labour_fm          edu.exp           life.exp      
##  Min.   :-2.8189   Min.   :-2.6247   Min.   :-2.7378   Min.   :-2.7188  
##  1st Qu.:-0.5233   1st Qu.:-0.5484   1st Qu.:-0.6782   1st Qu.:-0.6425  
##  Median : 0.3503   Median : 0.2316   Median : 0.1140   Median : 0.3056  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.5958   3rd Qu.: 0.7350   3rd Qu.: 0.7126   3rd Qu.: 0.6717  
##  Max.   : 2.6646   Max.   : 1.6632   Max.   : 2.4730   Max.   : 1.4218  
##       gni          maternal.mortality.ratio   teen.birth          per.pa       
##  Min.   :-0.9193   Min.   :-0.6992          Min.   :-1.1325   Min.   :-1.8203  
##  1st Qu.:-0.7243   1st Qu.:-0.6496          1st Qu.:-0.8394   1st Qu.:-0.7409  
##  Median :-0.3013   Median :-0.4726          Median :-0.3298   Median :-0.1403  
##  Mean   : 0.0000   Mean   : 0.0000          Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.3712   3rd Qu.: 0.1932          3rd Qu.: 0.6030   3rd Qu.: 0.6127  
##  Max.   : 5.6890   Max.   : 4.4899          Max.   : 3.8344   Max.   : 3.1850
# perform principal component analysis (with the SVD method)
pca_human <- prcomp(human)
pca_human_std <- prcomp(human_std)
# print out summaries of the standardized variables
summary(pca_human_std)
## Importance of components:
##                          PC1    PC2     PC3     PC4     PC5     PC6     PC7
## Standard deviation     2.071 1.1397 0.87505 0.77886 0.66196 0.53631 0.45900
## Proportion of Variance 0.536 0.1624 0.09571 0.07583 0.05477 0.03595 0.02634
## Cumulative Proportion  0.536 0.6984 0.79413 0.86996 0.92473 0.96069 0.98702
##                            PC8
## Standard deviation     0.32224
## Proportion of Variance 0.01298
## Cumulative Proportion  1.00000
summary(pca_human)
## Importance of components:
##                              PC1      PC2   PC3   PC4   PC5   PC6    PC7    PC8
## Standard deviation     1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912 0.1591
## Proportion of Variance 9.999e-01   0.0001  0.00  0.00 0.000 0.000 0.0000 0.0000
## Cumulative Proportion  9.999e-01   1.0000  1.00  1.00 1.000 1.000 1.0000 1.0000
# draw a biplot of the principal component representation and the original variables
biplot(pca_human, 
       choices = 1:2, 
       cex = c(0.8, 1), 
       col = c("grey40", "deeppink2"))
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length =
## arrow.len): zero-length arrow is of indeterminate angle and so skipped

biplot(pca_human_std, 
       choices = 1:2, 
       cex = c(0.8, 1), 
       col = c("grey40", "deeppink2"))

4.Give personal interpretations of the first two principal component dimensions based on the biplot drawn after PCA on the standardized human data

In this figure, the countries on the left are more concentrated, with an increase in educational expectations, life expectancy and the level of primary education attained by both men and women in this region, with a number of European or East Asian countries. In contrast, the number of women in parliament and the ratio of men to women in the labour market are concentrated at the top of the graph, mostly in the Nordic countries. Finally, the high maternal mortality and adolescent birth rates point to the countries on the right, which are mostly African countries.

  1. Tea data
#read data
tea <- read.csv("https://raw.githubusercontent.com/KimmoVehkalahti/Helsinki-Open-Data-Science/master/datasets/tea.csv", stringsAsFactors = TRUE)
#explore the data
str(tea); dim(tea)
## 'data.frame':    300 obs. of  36 variables:
##  $ breakfast       : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
##  $ tea.time        : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
##  $ evening         : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
##  $ lunch           : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
##  $ dinner          : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
##  $ always          : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
##  $ home            : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
##  $ work            : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
##  $ tearoom         : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
##  $ friends         : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
##  $ resto           : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
##  $ pub             : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
##  $ Tea             : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How             : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ sugar           : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ how             : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ where           : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ price           : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
##  $ age             : int  39 45 47 23 48 21 37 36 40 37 ...
##  $ sex             : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
##  $ SPC             : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
##  $ Sport           : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
##  $ age_Q           : Factor w/ 5 levels "+60","15-24",..: 4 5 5 2 5 2 4 4 4 4 ...
##  $ frequency       : Factor w/ 4 levels "+2/day","1 to 2/week",..: 3 3 1 3 1 3 4 2 1 1 ...
##  $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
##  $ spirituality    : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
##  $ healthy         : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
##  $ diuretic        : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
##  $ friendliness    : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
##  $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
##  $ feminine        : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
##  $ sophisticated   : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
##  $ slimming        : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ exciting        : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
##  $ relaxing        : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
##  $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...
## [1] 300  36

Tea dataset has 300 observations and 36 varibales, and all the variables are categorical.

#convert all variables to factors
tea$Tea <- factor(tea$Tea)
tea$How <- factor(tea$How)
tea$how <- factor(tea$how)
tea$sugar <- factor(tea$sugar)
tea$where <- factor(tea$where)
tea$lunch <- factor(tea$lunch)
library(dplyr)
library(tidyr)
# column names to keep in the dataset
keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")

# select the 'keep_columns' to create a new dataset
tea_time <- dplyr::select(tea, keep_columns)
## Warning: Using an external vector in selections was deprecated in tidyselect 1.1.0.
## ℹ Please use `all_of()` or `any_of()` instead.
##   # Was:
##   data %>% select(keep_columns)
## 
##   # Now:
##   data %>% select(all_of(keep_columns))
## 
## See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
# look at the summaries and structure of the data
summary(tea_time)
##         Tea         How                      how           sugar    
##  black    : 74   alone:195   tea bag           :170   No.sugar:155  
##  Earl Grey:193   lemon: 33   tea bag+unpackaged: 94   sugar   :145  
##  green    : 33   milk : 63   unpackaged        : 36                 
##                  other:  9                                          
##                   where           lunch    
##  chain store         :192   lunch    : 44  
##  chain store+tea shop: 78   Not.lunch:256  
##  tea shop            : 30                  
## 
str(tea_time)
## 'data.frame':    300 obs. of  6 variables:
##  $ Tea  : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
##  $ How  : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
##  $ how  : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
##  $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
##  $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
# visualize the dataset
library(ggplot2)
pivot_longer(tea_time, cols = everything()) %>% 
  ggplot(aes(value)) + 
  facet_wrap("name", scales = "free") +
  geom_bar() +
  theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))

In tea dataset, I choose 6 variables, The frequency distribution chart gives a very visual indication of the number of times each of the six variables was answered.

# multiple correspondence analysis
library(FactoMineR)
mca <- MCA(tea_time, graph = FALSE)

# summary of the model
summary(mca)
## 
## Call:
## MCA(X = tea_time, graph = FALSE) 
## 
## 
## Eigenvalues
##                        Dim.1   Dim.2   Dim.3   Dim.4   Dim.5   Dim.6   Dim.7
## Variance               0.279   0.261   0.219   0.189   0.177   0.156   0.144
## % of var.             15.238  14.232  11.964  10.333   9.667   8.519   7.841
## Cumulative % of var.  15.238  29.471  41.435  51.768  61.434  69.953  77.794
##                        Dim.8   Dim.9  Dim.10  Dim.11
## Variance               0.141   0.117   0.087   0.062
## % of var.              7.705   6.392   4.724   3.385
## Cumulative % of var.  85.500  91.891  96.615 100.000
## 
## Individuals (the 10 first)
##                       Dim.1    ctr   cos2    Dim.2    ctr   cos2    Dim.3
## 1                  | -0.298  0.106  0.086 | -0.328  0.137  0.105 | -0.327
## 2                  | -0.237  0.067  0.036 | -0.136  0.024  0.012 | -0.695
## 3                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 4                  | -0.530  0.335  0.460 | -0.318  0.129  0.166 |  0.211
## 5                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 6                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 7                  | -0.369  0.162  0.231 | -0.300  0.115  0.153 | -0.202
## 8                  | -0.237  0.067  0.036 | -0.136  0.024  0.012 | -0.695
## 9                  |  0.143  0.024  0.012 |  0.871  0.969  0.435 | -0.067
## 10                 |  0.476  0.271  0.140 |  0.687  0.604  0.291 | -0.650
##                       ctr   cos2  
## 1                   0.163  0.104 |
## 2                   0.735  0.314 |
## 3                   0.062  0.069 |
## 4                   0.068  0.073 |
## 5                   0.062  0.069 |
## 6                   0.062  0.069 |
## 7                   0.062  0.069 |
## 8                   0.735  0.314 |
## 9                   0.007  0.003 |
## 10                  0.643  0.261 |
## 
## Categories (the 10 first)
##                        Dim.1     ctr    cos2  v.test     Dim.2     ctr    cos2
## black              |   0.473   3.288   0.073   4.677 |   0.094   0.139   0.003
## Earl Grey          |  -0.264   2.680   0.126  -6.137 |   0.123   0.626   0.027
## green              |   0.486   1.547   0.029   2.952 |  -0.933   6.111   0.107
## alone              |  -0.018   0.012   0.001  -0.418 |  -0.262   2.841   0.127
## lemon              |   0.669   2.938   0.055   4.068 |   0.531   1.979   0.035
## milk               |  -0.337   1.420   0.030  -3.002 |   0.272   0.990   0.020
## other              |   0.288   0.148   0.003   0.876 |   1.820   6.347   0.102
## tea bag            |  -0.608  12.499   0.483 -12.023 |  -0.351   4.459   0.161
## tea bag+unpackaged |   0.350   2.289   0.056   4.088 |   1.024  20.968   0.478
## unpackaged         |   1.958  27.432   0.523  12.499 |  -1.015   7.898   0.141
##                     v.test     Dim.3     ctr    cos2  v.test  
## black                0.929 |  -1.081  21.888   0.382 -10.692 |
## Earl Grey            2.867 |   0.433   9.160   0.338  10.053 |
## green               -5.669 |  -0.108   0.098   0.001  -0.659 |
## alone               -6.164 |  -0.113   0.627   0.024  -2.655 |
## lemon                3.226 |   1.329  14.771   0.218   8.081 |
## milk                 2.422 |   0.013   0.003   0.000   0.116 |
## other                5.534 |  -2.524  14.526   0.197  -7.676 |
## tea bag             -6.941 |  -0.065   0.183   0.006  -1.287 |
## tea bag+unpackaged  11.956 |   0.019   0.009   0.000   0.226 |
## unpackaged          -6.482 |   0.257   0.602   0.009   1.640 |
## 
## Categorical variables (eta2)
##                      Dim.1 Dim.2 Dim.3  
## Tea                | 0.126 0.108 0.410 |
## How                | 0.076 0.190 0.394 |
## how                | 0.708 0.522 0.010 |
## sugar              | 0.065 0.001 0.336 |
## where              | 0.702 0.681 0.055 |
## lunch              | 0.000 0.064 0.111 |
# visualize MCA
plot(mca, invisible=c("ind"), graph.type = "classic", habillage = "quali")

In the Mac factor map, different colours indicate different variables. Red indicates how the tea is drunk, with lemon, milk, other or nothing. Green indicates how the tea is packaged. Pink indicates where the tea was purchased. Yellow indicates whether the tea was consumed before or after lunch. Black indicates the variety of tea. Blue indicates whether or not sugar was added. In the map we can observe that Earl Grey tea is usually served with milk and sugar and is usually served at breakfast time; black tea is served with lemon; green tea is usually bought unpackaged from a tea shop and is usually served at breakfast or after lunch.


Exercise 1 In this exercise, I will use the RATS dataset, and implement the analyses of Chapter 8 of MABS. Prepare the packages

library(dplyr)
library(tidyr)

read the RATS data set

RATS <- read.table("data/ratsl.txt", sep = ",", header = T)

RATS$ID <- factor(RATS$ID)
RATS$Group <- factor(RATS$Group)

glimpse(RATS)
## Rows: 176
## Columns: 5
## $ ID     <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3,…
## $ Group  <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1, 1, …
## $ WD     <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", …
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 555, 470…
## $ Time   <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, 8, …

The dataset has 176 rows and 5 columns.

# Plot the RATSL data
library(ggplot2)
ggplot(RATS, aes(x = Time, y = Weight, group = ID)) +
  geom_line(aes(linetype = Group))+
  scale_x_continuous(name = "Time(days)", breaks = seq(0, 60, 10))+
  scale_y_continuous(name = "Weight (grams)")+
  theme(legend.position = "top")

In the visualisation, it can be seen very visually that the weight of each rat increases slightly with time. Next the weights were standardised and the search for the presence of this change continued.

library(dplyr)
library(tidyr)
# Standardise the variable bprs
RATS_std <- RATS %>%
  group_by(ID) %>%
  mutate(stdweight = (Weight - mean(Weight))/sd(Weight)) %>%
  ungroup()

# Glimpse the data
glimpse(RATS_std)
## Rows: 176
## Columns: 6
## $ ID        <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2,…
## $ Group     <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1, …
## $ WD        <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1…
## $ Weight    <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 555, …
## $ Time      <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, …
## $ stdweight <dbl> -2.02487690, -1.73484555, -1.76767584, -0.90187616, -1.63589…
# Plot again with the standardised bprs
library(ggplot2)
ggplot(RATS_std, aes(x = Time, y = stdweight, linetype = Group)) +
  geom_line() +
  scale_x_continuous(name = "Time (days)", breaks = seq(0, 64, 7)) +
  facet_grid(. ~ Group, labeller = label_both) +
  scale_y_continuous(name = "standardized weight") 

After standardizing the data, it is easy to see that the weighting increases gradually.

Then summary measure analysis of the data set

# Number of subjects (per group):
n <- 11

library(dplyr)
library(tidyr)

# Summary data with mean and standard error of Weight by Group and Time 
RATSS <- RATS %>%
  group_by(Group, Time) %>%
  summarise( mean = mean(Weight), se = sd(Weight)/sqrt(n) ) %>%
  ungroup()
## `summarise()` has grouped output by 'Group'. You can override using the
## `.groups` argument.
# Glimpse the data
glimpse(RATSS)
## Rows: 33
## Columns: 4
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2…
## $ Time  <int> 1, 8, 15, 22, 29, 36, 43, 44, 50, 57, 64, 1, 8, 15, 22, 29, 36, …
## $ mean  <dbl> 250.625, 255.000, 254.375, 261.875, 264.625, 265.000, 267.375, 2…
## $ se    <dbl> 4.589478, 3.947710, 3.460116, 4.100800, 3.333956, 3.552939, 3.30…
# Plot the mean profiles
library(ggplot2)
ggplot(RATSS, aes(x = Time, y = mean, col = Group)) +
  geom_line() +
  #geom_errorbar(aes(ymin=mean-se, ymax=mean+se), width=0.3) +
  theme(legend.position = "right") +
  scale_y_continuous(name = "mean(Weight) +/- se(Weight)") +
  scale_x_continuous(name = "Time (days)", breaks = seq(1, 64, 7))

Applying the summary measure approach

library(dplyr)
library(tidyr)
# Summary data with mean and standard error of rats by treatment and week 
RATSS1 <- RATS %>%
  group_by(Group, ID) %>%
  summarise( mean=mean(Weight) ) %>%
  ungroup()
## `summarise()` has grouped output by 'Group'. You can override using the
## `.groups` argument.
# Glimpse the data
glimpse(RATSS)
## Rows: 33
## Columns: 4
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2…
## $ Time  <int> 1, 8, 15, 22, 29, 36, 43, 44, 50, 57, 64, 1, 8, 15, 22, 29, 36, …
## $ mean  <dbl> 250.625, 255.000, 254.375, 261.875, 264.625, 265.000, 267.375, 2…
## $ se    <dbl> 4.589478, 3.947710, 3.460116, 4.100800, 3.333956, 3.552939, 3.30…
# Draw a boxplot of the mean versus treatment
library(ggplot2)
ggplot(RATSS1, aes(x = Group, y = mean)) +
  geom_boxplot() +
  stat_summary(fun = "mean", geom = "point", shape=23, size=4, fill = "white") +
  scale_y_continuous(name = "mean(weight), days 1-64")

# Create a new data by filtering the outlier and adjust the ggplot code the draw the plot again with the new data
RATSS2 <- RATSS %>%
  filter(mean < 580)

library(ggplot2)
ggplot(RATSS2, aes(x = Group, y = mean))+
  geom_boxplot()+
  stat_summary(fun = "mean", geom = "point", shape = 23, size = 4, fill = "white")+
  scale_y_continuous(name = "mean(weight), days 1-64")

Anova

library(dplyr)
library(tidyr)
# Add the baseline from the original data as a new variable to the summary data
RATSS3 <- RATSS %>%
  mutate(baseline = RATS$day0)

# Fit the linear model with the mean as the response 
fit <- lm(mean ~ Group, data = RATSS3)

# Compute the analysis of variance table for the fitted model with anova()
anova(fit)
## Analysis of Variance Table
## 
## Response: mean
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## Group      2 437104  218552  1006.3 < 2.2e-16 ***
## Residuals 30   6516     217                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Exercise 2 In this exercise, I will use BPRS dataset. The explain of BPRS from MABS4IODS:“Here 40 male subjects were randomly assigned to one of two treatment groups and each subject was rated on the brief psychiatric rating scale (BPRS) measured before treatment be- gan (week 0) and then at weekly intervals for eight weeks. The BPRS assesses the level of 18 symptom constructs such as hostility, suspiciousness, halluci- nations and grandiosity; each of these is rated from one (not present) to seven (extremely severe). The scale is used to evaluate patients suspected of having schizophrenia.”

read the data set

BPRS <- read.table("data/bprsl.txt", sep = ",",  header = T)
# Access the packages dplyr and tidyr
library(dplyr)
library(tidyr)

# Extract the week number
BPRS <-  BPRS %>% 
            mutate(week = as.integer(substr(BPRS$weeks, 5, 5)))

# Take a glimpse at the BPRSL data
glimpse(BPRS)
## Rows: 360
## Columns: 5
## $ treatment <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …
## $ subject   <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1…
## $ weeks     <chr> "week0", "week0", "week0", "week0", "week0", "week0", "week0…
## $ bprs      <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, 38, 66, …
## $ week      <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, …
# Plot the data
library(ggplot2)
ggplot(BPRS, aes(x = week, y = bprs, group = subject)) +
  geom_line() +
  theme(legend.position = "top") +
  facet_grid(. ~ treatment, labeller = label_both)

Holding on to independence: The Linear model

# create a regression model BPRS_reg
BPRS_reg <- lm(bprs ~ week + treatment, data = BPRS)

# print out a summary of the model
summary(BPRS_reg)
## 
## Call:
## lm(formula = bprs ~ week + treatment, data = BPRS)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -22.454  -8.965  -3.196   7.002  50.244 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  45.8817     2.2949  19.993   <2e-16 ***
## week         -2.2704     0.2524  -8.995   <2e-16 ***
## treatment     0.5722     1.3034   0.439    0.661    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 12.37 on 357 degrees of freedom
## Multiple R-squared:  0.1851, Adjusted R-squared:  0.1806 
## F-statistic: 40.55 on 2 and 357 DF,  p-value: < 2.2e-16

The Random Intercept Model

# access library lme4
library(lme4)
## Loading required package: Matrix
## 
## Attaching package: 'Matrix'
## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack
# Create a random intercept model
BPRS_ref <- lmer(bprs ~ week + treatment + (week | subject), data = BPRS, REML = FALSE)

# Print the summary of the model
summary(BPRS_ref)
## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject)
##    Data: BPRS
## 
##      AIC      BIC   logLik deviance df.resid 
##   2745.4   2772.6  -1365.7   2731.4      353 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -2.8919 -0.6194 -0.0691  0.5531  3.7976 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev. Corr 
##  subject  (Intercept) 64.8222  8.0512        
##           week         0.9609  0.9802   -0.51
##  Residual             97.4305  9.8707        
## Number of obs: 360, groups:  subject, 20
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  45.8817     2.5685  17.863
## week         -2.2704     0.2977  -7.626
## treatment     0.5722     1.0405   0.550
## 
## Correlation of Fixed Effects:
##           (Intr) week  
## week      -0.477       
## treatment -0.608  0.000

Slippery slopes: Random Intercept and Random Slope Model

# create a random intercept and random slope model
library(lme4)
BPRS_ref1 <- lmer(bprs ~ week + treatment + (week | subject), data = BPRS, REML = FALSE)

# print a summary of the model
summary(BPRS_ref1)
## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject)
##    Data: BPRS
## 
##      AIC      BIC   logLik deviance df.resid 
##   2745.4   2772.6  -1365.7   2731.4      353 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -2.8919 -0.6194 -0.0691  0.5531  3.7976 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev. Corr 
##  subject  (Intercept) 64.8222  8.0512        
##           week         0.9609  0.9802   -0.51
##  Residual             97.4305  9.8707        
## Number of obs: 360, groups:  subject, 20
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  45.8817     2.5685  17.863
## week         -2.2704     0.2977  -7.626
## treatment     0.5722     1.0405   0.550
## 
## Correlation of Fixed Effects:
##           (Intr) week  
## week      -0.477       
## treatment -0.608  0.000
# perform an ANOVA test on the two models
anova(BPRS_ref1, BPRS_ref)
## Data: BPRS
## Models:
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## BPRS_ref: bprs ~ week + treatment + (week | subject)
##           npar    AIC    BIC  logLik deviance Chisq Df Pr(>Chisq)
## BPRS_ref1    7 2745.4 2772.6 -1365.7   2731.4                    
## BPRS_ref     7 2745.4 2772.6 -1365.7   2731.4     0  0

Time to interact: Random Intercept and Random Slope Model with interaction

# create a random intercept and random slope model with the interaction
library(lme4)
BPRS_ref2 <- lmer(bprs ~ week + treatment + (week | subject) + week * treatment, data = BPRS, REML = FALSE)

# print a summary of the model
summary(BPRS_ref2)
## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject) + week * treatment
##    Data: BPRS
## 
##      AIC      BIC   logLik deviance df.resid 
##   2744.3   2775.4  -1364.1   2728.3      352 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.0513 -0.6271 -0.0768  0.5288  3.9260 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev. Corr 
##  subject  (Intercept) 65.0049  8.0626        
##           week         0.9687  0.9842   -0.51
##  Residual             96.4702  9.8219        
## Number of obs: 360, groups:  subject, 20
## 
## Fixed effects:
##                Estimate Std. Error t value
## (Intercept)     50.1767     3.5159  14.271
## week            -3.3442     0.6711  -4.983
## treatment       -2.2911     1.9090  -1.200
## week:treatment   0.7158     0.4010   1.785
## 
## Correlation of Fixed Effects:
##             (Intr) week   trtmnt
## week        -0.768              
## treatment   -0.814  0.753       
## week:trtmnt  0.684 -0.896 -0.840
# perform an ANOVA test on the two models
anova(BPRS_ref2, BPRS_ref1) 
## Data: BPRS
## Models:
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## BPRS_ref2: bprs ~ week + treatment + (week | subject) + week * treatment
##           npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)  
## BPRS_ref1    7 2745.4 2772.6 -1365.7   2731.4                       
## BPRS_ref2    8 2744.3 2775.4 -1364.1   2728.3 3.1712  1    0.07495 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# draw the plot of BPRS with the observed Weight values
library(ggplot2)
ggplot(BPRS, aes(x = week, y = bprs, group = subject)) +
  geom_line(aes()) +
  scale_x_continuous(name = "Time (weeks)") +
  scale_y_continuous(name = "Observed BPRS") +
  theme(legend.position = "top") 

# Create a vector of the fitted values
Fitted <- fitted(BPRS_ref2)

library(dplyr)
library(tidyr)
# Create a new column fitted to RATSL
BPRS <- BPRS %>%
  mutate(Fitted = Fitted)

# draw the plot of RATSL with the Fitted values of weight
library(ggplot2)
ggplot(BPRS, aes(x = week, y = Fitted, group = subject)) +
  geom_line(aes()) +
  scale_x_continuous(name = "Time (weeks)", breaks = seq(0, 60, 20)) +
  scale_y_continuous(name = "Fitted BPRS") +
  theme(legend.position = "top")